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Matematiksel İstatistik Çalışma Soruları<=
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</head>
<body lang=3DTR style=3D'tab-interval:35.4pt'>
<div class=3DSection1>
<h1>MATEMATİKSEL İSTATİSTİK ÇALIŞMA SORULAR=
I</h1>
<p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><b style=3D=
'mso-bidi-font-weight:
normal'>TANIMLAR<o:p></o:p></b></p>
<p class=3DMsoNormal style=3D'margin-left:25.5pt;text-indent:-25.5pt;line-h=
eight:
150%;mso-list:l13 level1 lfo1;tab-stops:list 25.5pt'><![if !supportLists]><=
span
style=3D'mso-list:Ignore'>1.<span style=3D'font:7.0pt "Times New Roman"'>&n=
bsp;
</span></span><![endif]>Tesadüfi değişken nedir?</p>
<p class=3DMsoNormal style=3D'margin-left:25.5pt;text-indent:-25.5pt;line-h=
eight:
150%;mso-list:l13 level1 lfo1;tab-stops:list 25.5pt'><![if !supportLists]><=
span
style=3D'mso-list:Ignore'>2.<span style=3D'font:7.0pt "Times New Roman"'>&n=
bsp;
</span></span><![endif]>Dağılım fonksiyonu ne demektir <u>k&=
#305;saca</u>
açıklayınız ve olasılık <span class=3DSpell=
E>notasyonunu</span>
kullanarak gösteriniz.</p>
<p class=3DMsoNormal style=3D'margin-left:25.5pt;text-indent:-25.5pt;line-h=
eight:
150%;mso-list:l13 level1 lfo1;tab-stops:list 25.5pt'><![if !supportLists]><=
span
style=3D'mso-list:Ignore'>3.<span style=3D'font:7.0pt "Times New Roman"'>&n=
bsp;
</span></span><![endif]>Koşullu bir olasılık yoğunluk
fonksiyonunun özelliklerini maddeler halinde yazınız.</p>
<p class=3DMsoNormal style=3D'margin-left:25.5pt;text-indent:-25.5pt;line-h=
eight:
150%;mso-list:l13 level1 lfo1;tab-stops:list 25.5pt'><![if !supportLists]><=
span
style=3D'mso-list:Ignore'>4.<span style=3D'font:7.0pt "Times New Roman"'>&n=
bsp;
</span></span><![endif]>Koşullu bir dağılım fonksiyonun=
un
özelliklerini maddeler halinde yazınız.</p>
<p class=3DMsoNormal style=3D'margin-left:25.5pt;text-indent:-25.5pt;line-h=
eight:
150%;mso-list:l13 level1 lfo1;tab-stops:list 25.5pt'><![if !supportLists]><=
span
style=3D'mso-list:Ignore'>5.<span style=3D'font:7.0pt "Times New Roman"'>&n=
bsp;
</span></span><![endif]>İki boyutlu kesikli bir dağılım
fonksiyonunun özelliklerini maddeler halinde yazınız.</p>
<p class=3DMsoNormal style=3D'margin-left:25.5pt;text-indent:-25.5pt;line-h=
eight:
150%;mso-list:l13 level1 lfo1;tab-stops:list 25.5pt'><![if !supportLists]><=
span
style=3D'mso-list:Ignore'>6.<span style=3D'font:7.0pt "Times New Roman"'>&n=
bsp;
</span></span><![endif]>İki boyutlu olasılık yoğunluk
fonksiyonunun özelliklerini maddeler halinde yazınız.</p>
<p class=3DMsoNormal style=3D'margin-left:25.5pt;text-indent:-25.5pt;line-h=
eight:
150%;mso-list:l13 level1 lfo1;tab-stops:list 25.5pt'><![if !supportLists]><=
span
style=3D'mso-list:Ignore'>7.<span style=3D'font:7.0pt "Times New Roman"'>&n=
bsp;
</span></span><![endif]>İstatistikte kullanılan momentleri
istatistikte kullanım amacı ile birlikte bir tablo halinde
gösteriniz.</p>
<h1>TEMEL FONKSİYONLAR VE OLASILIK</h1>
<p class=3DMsoNormal style=3D'margin-left:25.5pt;text-indent:-25.5pt;line-h=
eight:
150%;mso-list:l13 level1 lfo1;tab-stops:list 25.5pt'><![if !supportLists]><=
span
style=3D'mso-list:Ignore'>8.<span style=3D'font:7.0pt "Times New Roman"'>&n=
bsp;
</span></span><![endif]>3 tane madeni paranın bir kez atılmas =
5;
deneyinde X, tura gelme sayısı iken, <span class=3DSpellE>X’=
;in</span>
olasılık ve birikimli olasılık fonksiyonunu
yazınız. P(X<=3D2)=3D?</p>
<p class=3DMsoNormal style=3D'margin-left:25.5pt;text-indent:-25.5pt;line-h=
eight:
150%;mso-list:l13 level1 lfo1;tab-stops:list 25.5pt'><![if !supportLists]><=
span
style=3D'mso-list:Ignore'>9.<span style=3D'font:7.0pt "Times New Roman"'>&n=
bsp;
</span></span><![endif]>2 tane hilesiz zarın bir kez atılması
deneyinde X, zarlar üzerindeki noktaların toplamı iken, <span
class=3DSpellE>X’in</span> olasılık ve birikimli
olasılık fonksiyonlarını yazınız. P(X>8)=
=3D?</p>
<p class=3DMsoNormal style=3D'margin-left:25.5pt;text-indent:-25.5pt;line-h=
eight:
150%;mso-list:l13 level1 lfo1;tab-stops:list 25.5pt'><![if !supportLists]><=
span
style=3D'mso-list:Ignore'>10.<span style=3D'font:7.0pt "Times New Roman"'>&=
nbsp;
</span></span><![endif]>1 madeni paranın ve bir zarın bir kez
atılması deneyinde X, zar üzerindeki noktaların
sayısı, ve Y madeni paranın yazı gelmesi sayıs =
5;
iken, (X,Y) iki boyutlu tesadüfi değişkeninin
dağılım fonksiyonunu yazınız. P(X>3,Y<1)=3D?=
</p>
<p class=3DMsoNormal style=3D'margin-left:25.5pt;text-indent:-25.5pt;line-h=
eight:
150%;mso-list:l13 level1 lfo1;tab-stops:list 25.5pt'><![if !supportLists]><=
span
style=3D'mso-list:Ignore'>11.<span style=3D'font:7.0pt "Times New Roman"'>&=
nbsp;
</span></span><![endif]>Kanser hastalığının var olup
olmadığını ortaya koyacak bir test
geliştirilmiştir. Bu test 0.05 olasılıkla yanlı=
51;
sonuç vermektedir. İçlerinde 0.005 oranında kanserli
hasta bulunan çok sayıda kişiye bu test
uygulanmıştır. Test sonucu kanser hastası olarak belirl=
enen
bir kişinin gerçekte kanserli olması
olasılığı nedir?</p>
<p class=3DMsoNormal style=3D'margin-left:25.5pt;text-indent:-25.5pt;line-h=
eight:
150%;mso-list:l13 level1 lfo1;tab-stops:list 25.5pt'><![if !supportLists]><=
span
style=3D'mso-list:Ignore'>12.<span style=3D'font:7.0pt "Times New Roman"'>&=
nbsp;
</span></span><![endif]>f(x)=3Da+bx<sup>3</sup>, 0<span style=3D'font-famil=
y:Symbol;
mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:"Times New Ro=
man";
mso-char-type:symbol;mso-symbol-font-family:Symbol'><span style=3D'mso-char=
-type:
symbol;mso-symbol-font-family:Symbol'>£</span></span>x<span
style=3D'font-family:Symbol;mso-ascii-font-family:"Times New Roman";mso-han=
si-font-family:
"Times New Roman";mso-char-type:symbol;mso-symbol-font-family:Symbol'><span
style=3D'mso-char-type:symbol;mso-symbol-font-family:Symbol'>£</span>=
</span>1
dağılan X tesadüfi değişkeninin beklenen değe=
ri
E(X)=3D2/5 olduğuna göre a ve <span class=3DSpellE>b’nin</s=
pan>
değerlerini hesaplayınız.</p>
<p class=3DMsoNormal style=3D'margin-left:25.5pt;text-indent:-25.5pt;line-h=
eight:
150%;mso-list:l13 level1 lfo1;tab-stops:list 25.5pt'><![if !supportLists]><=
span
style=3D'mso-list:Ignore'>13.<span style=3D'font:7.0pt "Times New Roman"'>&=
nbsp;
</span></span><![endif]>f(x)=3Da+bx<sup>3</sup>, 0<span style=3D'font-famil=
y:Symbol;
mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:"Times New Ro=
man";
mso-char-type:symbol;mso-symbol-font-family:Symbol'><span style=3D'mso-char=
-type:
symbol;mso-symbol-font-family:Symbol'>£</span></span>x<span
style=3D'font-family:Symbol;mso-ascii-font-family:"Times New Roman";mso-han=
si-font-family:
"Times New Roman";mso-char-type:symbol;mso-symbol-font-family:Symbol'><span
style=3D'mso-char-type:symbol;mso-symbol-font-family:Symbol'>£</span>=
</span>1
dağılan X tesadüfi değişkeninin varyansı V(X)=
=3D4/3
olduğuna göre a ve <span class=3DSpellE>b’nin</span>
değerlerini hesaplayınız.</p>
<p class=3DMsoNormal style=3D'margin-left:25.5pt;text-indent:-25.5pt;line-h=
eight:
150%;mso-list:l13 level1 lfo1;tab-stops:list 25.5pt'><![if !supportLists]><=
span
style=3D'mso-list:Ignore'>14.<span style=3D'font:7.0pt "Times New Roman"'>&=
nbsp;
</span></span><![endif]>f(x)=3Dax<sup>3</sup>, 0<span style=3D'font-family:=
Symbol;
mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:"Times New Ro=
man";
mso-char-type:symbol;mso-symbol-font-family:Symbol'><span style=3D'mso-char=
-type:
symbol;mso-symbol-font-family:Symbol'>£</span></span>x<span
style=3D'font-family:Symbol;mso-ascii-font-family:"Times New Roman";mso-han=
si-font-family:
"Times New Roman";mso-char-type:symbol;mso-symbol-font-family:Symbol'><span
style=3D'mso-char-type:symbol;mso-symbol-font-family:Symbol'>£</span>=
</span>1
dağılan X tesadüfi değişkeninin varyansı V(X)=
=3D4/150
olduğuna göre <span class=3DSpellE>a’nın</span>
değerini hesaplayınız.</p>
<p class=3DMsoNormal style=3D'margin-left:25.5pt;text-indent:-25.5pt;line-h=
eight:
150%;mso-list:l13 level1 lfo1;tab-stops:list 25.5pt'><![if !supportLists]><=
span
style=3D'mso-list:Ignore'>15.<span style=3D'font:7.0pt "Times New Roman"'>&=
nbsp;
</span></span><![endif]>(X, Y) tesadüfi değişkeninin ortak
olasılık yoğunluk fonksiyonu f(x,y)=3De<sup>-(x+y)</sup>,x&g=
t;0,y>0
şeklinde verilmiştir. P(X+Y < a) =3D 0.5 olması iç=
in
“a” kaç olmalıdır?</p>
<p class=3DMsoNormal style=3D'margin-left:25.5pt;text-indent:-25.5pt;line-h=
eight:
150%;mso-list:l13 level1 lfo1;tab-stops:list 25.5pt'><![if !supportLists]><=
span
style=3D'mso-list:Ignore'>16.<span style=3D'font:7.0pt "Times New Roman"'>&=
nbsp;
</span></span><![endif]>Aşağıda verilen ortak
dağılım fonksiyonu için z(x/y) koşullu
olasılık yoğunluk fonksiyonunu belirleyiniz. <sub><!--[if gt=
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<v:f eqn=3D"sum @0 1 0"/>
<v:f eqn=3D"sum 0 0 @1"/>
<v:f eqn=3D"prod @2 1 2"/>
<v:f eqn=3D"prod @3 21600 pixelWidth"/>
<v:f eqn=3D"prod @3 21600 pixelHeight"/>
<v:f eqn=3D"sum @0 0 1"/>
<v:f eqn=3D"prod @6 1 2"/>
<v:f eqn=3D"prod @7 21600 pixelWidth"/>
<v:f eqn=3D"sum @8 21600 0"/>
<v:f eqn=3D"prod @7 21600 pixelHeight"/>
<v:f eqn=3D"sum @10 21600 0"/>
</v:formulas>
<v:path o:extrusionok=3D"f" gradientshapeok=3D"t" o:connecttype=3D"rect"/>
<o:lock v:ext=3D"edit" aspectratio=3D"t"/>
</v:shapetype><v:shape id=3D"_x0000_i1061" type=3D"#_x0000_t75" style=3D'wi=
dth:161.25pt;
height:51.75pt' o:ole=3D"">
<v:imagedata src=3D"matist_sorular_dosyalar/image001.wmz" o:title=3D""/>
</v:shape><![endif]--><![if !vml]><img width=3D215 height=3D69
src=3D"matist_sorular_dosyalar/image002.gif" v:shapes=3D"_x0000_i1061"><![e=
ndif]></sub><!--[if gte mso 9]><xml>
<o:OLEObject Type=3D"Embed" ProgID=3D"Equation.2" ShapeID=3D"_x0000_i1061"
DrawAspect=3D"Content" ObjectID=3D"_1198912089">
</o:OLEObject>
</xml><![endif]--></p>
<p class=3DMsoNormal style=3D'margin-left:25.5pt;text-indent:-25.5pt;line-h=
eight:
150%;mso-list:l13 level1 lfo1;tab-stops:list 25.5pt'><![if !supportLists]><=
span
style=3D'mso-list:Ignore'>17.<span style=3D'font:7.0pt "Times New Roman"'>&=
nbsp;
</span></span><![endif]>(X,Y) iki boyutlu sürekli tesadüfi
değişkeninin ortak olasılık yoğunluk fonksiyonu
aşağıda verilmiştir. g(y), H(x), k(x/y=3D1), F(x,y)
fonksiyonlarını bulunuz ve P(1<=3Dy<2,x<0.5)
hesaplayınız. X ve Y bağımlı olaylar mıdı=
;r?<br>
<sub><!--[if gte vml 1]><v:shape id=3D"_x0000_i1025" type=3D"#_x0000_t75" s=
tyle=3D'width:201.75pt;
height:48pt' o:ole=3D"">
<v:imagedata src=3D"matist_sorular_dosyalar/image003.wmz" o:title=3D""/>
</v:shape><![endif]--><![if !vml]><img width=3D269 height=3D64
src=3D"matist_sorular_dosyalar/image004.gif" v:shapes=3D"_x0000_i1025"><![e=
ndif]></sub><!--[if gte mso 9]><xml>
<o:OLEObject Type=3D"Embed" ProgID=3D"Equation.3" ShapeID=3D"_x0000_i1025"
DrawAspect=3D"Content" ObjectID=3D"_1198912090">
</o:OLEObject>
</xml><![endif]--></p>
<p class=3DMsoNormal style=3D'margin-left:25.5pt;text-indent:-25.5pt;line-h=
eight:
150%;mso-list:l13 level1 lfo1;tab-stops:list 25.5pt'><![if !supportLists]><=
span
style=3D'mso-list:Ignore'>18.<span style=3D'font:7.0pt "Times New Roman"'>&=
nbsp;
</span></span><![endif]>(X,Y) iki boyutlu sürekli tesadüfi
değişkeninin ortak olasılık yoğunluk fonksiyonu
aşağıda verilmiştir. g(y), H(x), k(x/1<span
style=3D'font-family:Symbol;mso-ascii-font-family:"Times New Roman";mso-han=
si-font-family:
"Times New Roman";mso-char-type:symbol;mso-symbol-font-family:Symbol'><span
style=3D'mso-char-type:symbol;mso-symbol-font-family:Symbol'>£</span>=
</span>y<4),
F(x,y) fonksiyonlarını bulunuz ve P(10<span style=3D'font-family:=
Symbol;
mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:"Times New Ro=
man";
mso-char-type:symbol;mso-symbol-font-family:Symbol'><span style=3D'mso-char=
-type:
symbol;mso-symbol-font-family:Symbol'>£</span></span>y,x<0.5)
hesaplayınız. X ve Y bağımlı olaylar mıdı=
;r?<br>
<sub><!--[if gte vml 1]><v:shape id=3D"_x0000_i1026" type=3D"#_x0000_t75" s=
tyle=3D'width:195.75pt;
height:45.75pt' o:ole=3D"">
<v:imagedata src=3D"matist_sorular_dosyalar/image005.wmz" o:title=3D""/>
</v:shape><![endif]--><![if !vml]><img width=3D261 height=3D61
src=3D"matist_sorular_dosyalar/image006.gif" v:shapes=3D"_x0000_i1026"><![e=
ndif]></sub><!--[if gte mso 9]><xml>
<o:OLEObject Type=3D"Embed" ProgID=3D"Equation.3" ShapeID=3D"_x0000_i1026"
DrawAspect=3D"Content" ObjectID=3D"_1198912091">
</o:OLEObject>
</xml><![endif]--></p>
<p class=3DMsoNormal style=3D'margin-left:25.5pt;text-indent:-25.5pt;line-h=
eight:
150%;mso-list:l13 level1 lfo1;tab-stops:list 25.5pt'><![if !supportLists]><=
span
style=3D'mso-list:Ignore'>19.<span style=3D'font:7.0pt "Times New Roman"'>&=
nbsp;
</span></span><![endif]>Ortak olasılık yoğunluk fonksiyonu
f(x,y)=3D3x, 0<span style=3D'font-family:Symbol;mso-ascii-font-family:"Time=
s New Roman";
mso-hansi-font-family:"Times New Roman";mso-char-type:symbol;mso-symbol-fon=
t-family:
Symbol'><span style=3D'mso-char-type:symbol;mso-symbol-font-family:Symbol'>=
£</span></span>y<span
style=3D'font-family:Symbol;mso-ascii-font-family:"Times New Roman";mso-han=
si-font-family:
"Times New Roman";mso-char-type:symbol;mso-symbol-font-family:Symbol'><span
style=3D'mso-char-type:symbol;mso-symbol-font-family:Symbol'>£</span>=
</span>x<span
style=3D'font-family:Symbol;mso-ascii-font-family:"Times New Roman";mso-han=
si-font-family:
"Times New Roman";mso-char-type:symbol;mso-symbol-font-family:Symbol'><span
style=3D'mso-char-type:symbol;mso-symbol-font-family:Symbol'>£</span>=
</span>1
olduğuna göre P(X+Y<1) olasılığını
hesaplayınız.</p>
<p class=3DMsoNormal style=3D'margin-left:25.5pt;text-indent:-25.5pt;line-h=
eight:
150%;mso-list:l13 level1 lfo1;tab-stops:list 25.5pt'><![if !supportLists]><=
span
style=3D'mso-list:Ignore'>20.<span style=3D'font:7.0pt "Times New Roman"'>&=
nbsp;
</span></span><![endif]>50 öğrenciden 15’i fizik, 20’=
si
kimya dersinden başarısızdır. 25 öğrenci her =
iki
dersten de başarısızdır. Fizik ve kimya derslerinden
başarısız olma olayları bağımsız olaylar
mıdır? Kanıtlayınız.</p>
<p class=3DMsoNormal style=3D'margin-left:25.5pt;text-indent:-25.5pt;line-h=
eight:
150%;mso-list:l13 level1 lfo1;tab-stops:list 25.5pt'><![if !supportLists]><=
span
style=3D'mso-list:Ignore'>21.<span style=3D'font:7.0pt "Times New Roman"'>&=
nbsp;
</span></span><![endif]>Aşağıda ortak olasılık
yoğunluk fonksiyonu verilen X ve Y tesadüfi değişkenleri
istatistiksel olarak bağımsız mıdır?
Kanıtlayınız. <sub><!--[if gte vml 1]><v:shape id=3D"_x0000_=
i1062"
type=3D"#_x0000_t75" style=3D'width:153.75pt;height:35.25pt' o:ole=3D"">
<v:imagedata src=3D"matist_sorular_dosyalar/image007.wmz" o:title=3D""/>
</v:shape><![endif]--><![if !vml]><img width=3D205 height=3D47
src=3D"matist_sorular_dosyalar/image008.gif" v:shapes=3D"_x0000_i1062"><![e=
ndif]></sub><!--[if gte mso 9]><xml>
<o:OLEObject Type=3D"Embed" ProgID=3D"Equation.2" ShapeID=3D"_x0000_i1062"
DrawAspect=3D"Content" ObjectID=3D"_1198912092">
</o:OLEObject>
</xml><![endif]--></p>
<p class=3DMsoNormal style=3D'margin-left:25.5pt;text-indent:-25.5pt;line-h=
eight:
150%;mso-list:l13 level1 lfo1;tab-stops:list 25.5pt'><![if !supportLists]><=
span
style=3D'mso-list:Ignore'>22.<span style=3D'font:7.0pt "Times New Roman"'>&=
nbsp;
</span></span><![endif]>Koşullu olasılık ve marjinal
olasılık yoğunluk fonksiyonları verilen X ve Y <span
class=3DSpellE>tesaadüfi</span> değişkenlerinin ortak
dağılım fonksiyonunu belirleyiniz. <sub><!--[if gte vml 1]><=
v:shape
id=3D"_x0000_i1063" type=3D"#_x0000_t75" style=3D'width:158.25pt;height:32=
.25pt'
o:ole=3D"">
<v:imagedata src=3D"matist_sorular_dosyalar/image009.wmz" o:title=3D""/>
</v:shape><![endif]--><![if !vml]><img width=3D211 height=3D43
src=3D"matist_sorular_dosyalar/image010.gif" v:shapes=3D"_x0000_i1063"><![e=
ndif]></sub><!--[if gte mso 9]><xml>
<o:OLEObject Type=3D"Embed" ProgID=3D"Equation.2" ShapeID=3D"_x0000_i1063"
DrawAspect=3D"Content" ObjectID=3D"_1198912093">
</o:OLEObject>
</xml><![endif]--></p>
<h1>BEKLENEN DEĞER, VARYANS</h1>
<p class=3DMsoNormal style=3D'margin-left:25.5pt;text-indent:-25.5pt;line-h=
eight:
150%;mso-list:l13 level1 lfo1;tab-stops:list 25.5pt'><![if !supportLists]><=
span
style=3D'mso-list:Ignore'>23.<span style=3D'font:7.0pt "Times New Roman"'>&=
nbsp;
</span></span><![endif]>Üniversiteye giriş için uygulanan
sınavın sayısal ve sözel bölümlerindeki
soruların bir öğrenci tarafından doğru cevaplanma
yüzdeleri sırasıyla (X) ve (Y) aşağıdaki
şekilde tesadüfi dağılmıştır. Sayıs=
al
bölümün yüzde 30ile 40’ı arasında
doğru cevaplayan bir öğrencinin sözel bölümde
doğru cevaplama yüzdesinin kaç olması beklenir?<br>
<sub><!--[if gte vml 1]><v:shape id=3D"_x0000_i1027" type=3D"#_x0000_t75" s=
tyle=3D'width:207pt;
height:48pt' o:ole=3D"">
<v:imagedata src=3D"matist_sorular_dosyalar/image011.wmz" o:title=3D""/>
</v:shape><![endif]--><![if !vml]><img width=3D276 height=3D64
src=3D"matist_sorular_dosyalar/image012.gif" v:shapes=3D"_x0000_i1027"><![e=
ndif]></sub><!--[if gte mso 9]><xml>
<o:OLEObject Type=3D"Embed" ProgID=3D"Equation.3" ShapeID=3D"_x0000_i1027"
DrawAspect=3D"Content" ObjectID=3D"_1198912094">
</o:OLEObject>
</xml><![endif]--></p>
<p class=3DMsoNormal style=3D'margin-left:25.5pt;text-indent:-25.5pt;line-h=
eight:
150%;mso-list:l13 level1 lfo1;tab-stops:list 25.5pt'><![if !supportLists]><=
span
style=3D'mso-list:Ignore'>24.<span style=3D'font:7.0pt "Times New Roman"'>&=
nbsp;
</span></span><![endif]>Bir satış personelinin günde
yaptığı benzin masrafı (X) ve buna karşıl =
5;k
şirketinden yapılan destekleme ödemesi (Y) iken, ortak
olasılık yoğunluk fonksiyonu aşağıda
verilmiştir. Satış personelinin 12 ile 15 TL arasında h=
arcama
yapması durumunda ne kadar destekleme ödemesi yapılması
beklenir?<br>
<sub><!--[if gte vml 1]><v:shape id=3D"_x0000_i1028" type=3D"#_x0000_t75" s=
tyle=3D'width:237pt;
height:50.25pt' o:ole=3D"">
<v:imagedata src=3D"matist_sorular_dosyalar/image013.wmz" o:title=3D""/>
</v:shape><![endif]--><![if !vml]><img width=3D316 height=3D67
src=3D"matist_sorular_dosyalar/image014.gif" v:shapes=3D"_x0000_i1028"><![e=
ndif]></sub><!--[if gte mso 9]><xml>
<o:OLEObject Type=3D"Embed" ProgID=3D"Equation.3" ShapeID=3D"_x0000_i1028"
DrawAspect=3D"Content" ObjectID=3D"_1198912095">
</o:OLEObject>
</xml><![endif]--></p>
<p class=3DMsoNormal style=3D'margin-left:25.5pt;text-indent:-25.5pt;line-h=
eight:
150%;mso-list:l13 level1 lfo1;tab-stops:list 25.5pt'><![if !supportLists]><=
span
style=3D'mso-list:Ignore'>25.<span style=3D'font:7.0pt "Times New Roman"'>&=
nbsp;
</span></span><![endif]>X tesadüfi değişkeninin karesinin
beklenen değerini hesaplayınız. <sub><!--[if gte vml 1]><v:s=
hape
id=3D"_x0000_i1029" type=3D"#_x0000_t75" style=3D'width:129pt;height:39.75=
pt' o:ole=3D"">
<v:imagedata src=3D"matist_sorular_dosyalar/image015.wmz" o:title=3D""/>
</v:shape><![endif]--><![if !vml]><img width=3D172 height=3D53
src=3D"matist_sorular_dosyalar/image016.gif" v:shapes=3D"_x0000_i1029"><![e=
ndif]></sub><!--[if gte mso 9]><xml>
<o:OLEObject Type=3D"Embed" ProgID=3D"Equation.3" ShapeID=3D"_x0000_i1029"
DrawAspect=3D"Content" ObjectID=3D"_1198912096">
</o:OLEObject>
</xml><![endif]--></p>
<p class=3DMsoNormal style=3D'margin-left:25.5pt;text-indent:-25.5pt;line-h=
eight:
150%;mso-list:l13 level1 lfo1;tab-stops:list 25.5pt'><![if !supportLists]><=
span
style=3D'mso-list:Ignore'>26.<span style=3D'font:7.0pt "Times New Roman"'>&=
nbsp;
</span></span><![endif]>X tesadüfi değişkeni için
E[(X-E(X))<sup>2</sup>] değerini hesaplayınız. <sub><!--[if =
gte vml 1]><v:shape
id=3D"_x0000_i1030" type=3D"#_x0000_t75" style=3D'width:198pt;height:48pt'=
o:ole=3D"">
<v:imagedata src=3D"matist_sorular_dosyalar/image017.wmz" o:title=3D""/>
</v:shape><![endif]--><![if !vml]><img width=3D264 height=3D64
src=3D"matist_sorular_dosyalar/image018.gif" v:shapes=3D"_x0000_i1030"><![e=
ndif]></sub><!--[if gte mso 9]><xml>
<o:OLEObject Type=3D"Embed" ProgID=3D"Equation.3" ShapeID=3D"_x0000_i1030"
DrawAspect=3D"Content" ObjectID=3D"_1198912097">
</o:OLEObject>
</xml><![endif]--></p>
<p class=3DMsoNormal style=3D'margin-left:25.5pt;text-indent:-25.5pt;line-h=
eight:
150%;mso-list:l13 level1 lfo1;tab-stops:list 25.5pt'><![if !supportLists]><=
span
style=3D'mso-list:Ignore'>27.<span style=3D'font:7.0pt "Times New Roman"'>&=
nbsp;
</span></span><![endif]><sub><!--[if gte vml 1]><v:shape id=3D"_x0000_i1031"
type=3D"#_x0000_t75" style=3D'width:294pt;height:36pt' o:ole=3D"">
<v:imagedata src=3D"matist_sorular_dosyalar/image019.wmz" o:title=3D""/>
</v:shape><![endif]--><![if !vml]><img width=3D392 height=3D48
src=3D"matist_sorular_dosyalar/image020.gif" v:shapes=3D"_x0000_i1031"><![e=
ndif]></sub><!--[if gte mso 9]><xml>
<o:OLEObject Type=3D"Embed" ProgID=3D"Equation.3" ShapeID=3D"_x0000_i1031"
DrawAspect=3D"Content" ObjectID=3D"_1198912098">
</o:OLEObject>
</xml><![endif]--></p>
<p class=3DMsoNormal style=3D'margin-left:25.5pt;text-indent:-25.5pt;line-h=
eight:
150%;mso-list:l13 level1 lfo1;tab-stops:list 25.5pt'><![if !supportLists]><=
span
style=3D'mso-list:Ignore'>28.<span style=3D'font:7.0pt "Times New Roman"'>&=
nbsp;
</span></span><![endif]>Ortak olasılık yoğunluk fonksiyonu
f(x,y)=3D3x, 0<span style=3D'font-family:Symbol;mso-ascii-font-family:"Time=
s New Roman";
mso-hansi-font-family:"Times New Roman";mso-char-type:symbol;mso-symbol-fon=
t-family:
Symbol'><span style=3D'mso-char-type:symbol;mso-symbol-font-family:Symbol'>=
£</span></span>y<span
style=3D'font-family:Symbol;mso-ascii-font-family:"Times New Roman";mso-han=
si-font-family:
"Times New Roman";mso-char-type:symbol;mso-symbol-font-family:Symbol'><span
style=3D'mso-char-type:symbol;mso-symbol-font-family:Symbol'>£</span>=
</span>x<span
style=3D'font-family:Symbol;mso-ascii-font-family:"Times New Roman";mso-han=
si-font-family:
"Times New Roman";mso-char-type:symbol;mso-symbol-font-family:Symbol'><span
style=3D'mso-char-type:symbol;mso-symbol-font-family:Symbol'>£</span>=
</span>1
olduğuna göre V(x/Y=3Dy) nedir?</p>
<p class=3DMsoNormal style=3D'margin-left:25.5pt;text-indent:-25.5pt;line-h=
eight:
150%;mso-list:l13 level1 lfo1;tab-stops:list 25.5pt'><![if !supportLists]><=
span
style=3D'mso-list:Ignore'>29.<span style=3D'font:7.0pt "Times New Roman"'>&=
nbsp;
</span></span><![endif]>Bir zar ve bir madeni paranın bir kez
atılması deneyinde X tesadüfi değişkeni zarın
üzerindeki noktaların sayısı, Y ise para
atıldığında ortaya çıkan yazı
sayısıdır. <span class=3DSpellE>Y’nin</span> X>3
için koşullu varyansını hesaplayınız.</p>
<p class=3DMsoNormal style=3D'margin-left:25.5pt;text-indent:-25.5pt;line-h=
eight:
150%;mso-list:l13 level1 lfo1;tab-stops:list 25.5pt'><![if !supportLists]><=
span
style=3D'mso-list:Ignore'>30.<span style=3D'font:7.0pt "Times New Roman"'>&=
nbsp;
</span></span><![endif]>(X, Y) iki boyutlu tesadüfi değişken
için V(Y/1<X<2) koşullu varyansını
hesaplayınız. <sub><!--[if gte vml 1]><v:shape id=3D"_x0000_i1032"
type=3D"#_x0000_t75" style=3D'width:171pt;height:48pt' o:ole=3D"">
<v:imagedata src=3D"matist_sorular_dosyalar/image021.wmz" o:title=3D""/>
</v:shape><![endif]--><![if !vml]><img width=3D228 height=3D64
src=3D"matist_sorular_dosyalar/image022.gif" v:shapes=3D"_x0000_i1032"><![e=
ndif]></sub><!--[if gte mso 9]><xml>
<o:OLEObject Type=3D"Embed" ProgID=3D"Equation.3" ShapeID=3D"_x0000_i1032"
DrawAspect=3D"Content" ObjectID=3D"_1198912099">
</o:OLEObject>
</xml><![endif]--></p>
<p class=3DMsoNormal style=3D'margin-left:25.5pt;text-indent:-25.5pt;line-h=
eight:
150%;mso-list:l13 level1 lfo1;tab-stops:list 25.5pt'><![if !supportLists]><=
span
style=3D'mso-list:Ignore'>31.<span style=3D'font:7.0pt "Times New Roman"'>&=
nbsp;
</span></span><![endif]><sub><!--[if gte vml 1]><v:shape id=3D"_x0000_i1033"
type=3D"#_x0000_t75" style=3D'width:311.25pt;height:36pt' o:ole=3D"">
<v:imagedata src=3D"matist_sorular_dosyalar/image023.wmz" o:title=3D""/>
</v:shape><![endif]--><![if !vml]><img width=3D415 height=3D48
src=3D"matist_sorular_dosyalar/image024.gif" v:shapes=3D"_x0000_i1033"><![e=
ndif]></sub><!--[if gte mso 9]><xml>
<o:OLEObject Type=3D"Embed" ProgID=3D"Equation.3" ShapeID=3D"_x0000_i1033"
DrawAspect=3D"Content" ObjectID=3D"_1198912100">
</o:OLEObject>
</xml><![endif]--></p>
<p class=3DMsoNormal style=3D'margin-left:25.5pt;text-indent:-25.5pt;line-h=
eight:
150%;mso-list:l13 level1 lfo1;tab-stops:list 25.5pt'><![if !supportLists]><=
span
style=3D'mso-list:Ignore'>32.<span style=3D'font:7.0pt "Times New Roman"'>&=
nbsp;
</span></span><![endif]>(X,Y) ortak olasılık yoğunluk fonksi=
yonu
<sub><!--[if gte vml 1]><v:shape id=3D"_x0000_i1034" type=3D"#_x0000_t75" s=
tyle=3D'width:231pt;
height:30.75pt' o:ole=3D"">
<v:imagedata src=3D"matist_sorular_dosyalar/image025.wmz" o:title=3D""/>
</v:shape><![endif]--><![if !vml]><img width=3D308 height=3D41
src=3D"matist_sorular_dosyalar/image026.gif" v:shapes=3D"_x0000_i1034"><![e=
ndif]></sub><!--[if gte mso 9]><xml>
<o:OLEObject Type=3D"Embed" ProgID=3D"Equation.3" ShapeID=3D"_x0000_i1034"
DrawAspect=3D"Content" ObjectID=3D"_1198912101">
</o:OLEObject>
</xml><![endif]--><span style=3D'mso-spacerun:yes'> </span>ise <b><i>r=
</i></b>
ilişki katsayısını hesaplayınız.</p>
<h1>BEKLENEN DEĞER VE ÖZELLİKLERİ</h1>
<p class=3DMsoNormal style=3D'margin-left:25.5pt;text-indent:-25.5pt;line-h=
eight:
150%;mso-list:l13 level1 lfo1;tab-stops:list 25.5pt'><![if !supportLists]><=
span
style=3D'mso-list:Ignore'>33.<span style=3D'font:7.0pt "Times New Roman"'>&=
nbsp;
</span></span><![endif]>3 tesadüfi değişkenin istatistikleri=
<span
style=3D'font-family:Symbol;mso-ascii-font-family:"Times New Roman";mso-han=
si-font-family:
"Times New Roman";mso-char-type:symbol;mso-symbol-font-family:Symbol'><span
style=3D'mso-char-type:symbol;mso-symbol-font-family:Symbol'>m</span></span=
><sub>C</sub>=3D75,
<span style=3D'font-family:Symbol;mso-ascii-font-family:"Times New Roman";
mso-hansi-font-family:"Times New Roman";mso-char-type:symbol;mso-symbol-fon=
t-family:
Symbol'><span style=3D'mso-char-type:symbol;mso-symbol-font-family:Symbol'>=
m</span></span><sub>I</sub>=3D20,
<span style=3D'font-family:Symbol;mso-ascii-font-family:"Times New Roman";
mso-hansi-font-family:"Times New Roman";mso-char-type:symbol;mso-symbol-fon=
t-family:
Symbol'><span style=3D'mso-char-type:symbol;mso-symbol-font-family:Symbol'>=
m</span></span><sub>G</sub>=3D5;
<span style=3D'font-family:Symbol;mso-ascii-font-family:"Times New Roman";
mso-hansi-font-family:"Times New Roman";mso-char-type:symbol;mso-symbol-fon=
t-family:
Symbol'><span style=3D'mso-char-type:symbol;mso-symbol-font-family:Symbol'>=
s</span></span><sub>C</sub>=3D10,
<span style=3D'font-family:Symbol;mso-ascii-font-family:"Times New Roman";
mso-hansi-font-family:"Times New Roman";mso-char-type:symbol;mso-symbol-fon=
t-family:
Symbol'><span style=3D'mso-char-type:symbol;mso-symbol-font-family:Symbol'>=
s</span></span><sub>I</sub>=3D5,
<span style=3D'font-family:Symbol;mso-ascii-font-family:"Times New Roman";
mso-hansi-font-family:"Times New Roman";mso-char-type:symbol;mso-symbol-fon=
t-family:
Symbol'><span style=3D'mso-char-type:symbol;mso-symbol-font-family:Symbol'>=
s</span></span><sub>G</sub>=3D1;
<span style=3D'font-family:Symbol;mso-ascii-font-family:"Times New Roman";
mso-hansi-font-family:"Times New Roman";mso-char-type:symbol;mso-symbol-fon=
t-family:
Symbol'><span style=3D'mso-char-type:symbol;mso-symbol-font-family:Symbol'>=
s</span></span><sub>CI</sub>=3D-50,
<span style=3D'font-family:Symbol;mso-ascii-font-family:"Times New Roman";
mso-hansi-font-family:"Times New Roman";mso-char-type:symbol;mso-symbol-fon=
t-family:
Symbol'><span style=3D'mso-char-type:symbol;mso-symbol-font-family:Symbol'>=
s</span></span><sub>CG</sub>=3D20,
<span style=3D'font-family:Symbol;mso-ascii-font-family:"Times New Roman";
mso-hansi-font-family:"Times New Roman";mso-char-type:symbol;mso-symbol-fon=
t-family:
Symbol'><span style=3D'mso-char-type:symbol;mso-symbol-font-family:Symbol'>=
s</span></span><sub>IG</sub>=3D
-10 ise Y=3DC+I+G tesadüfi değişkeninin varyansını
hesaplayınız.</p>
<p class=3DMsoNormal style=3D'margin-left:25.5pt;text-indent:-25.5pt;line-h=
eight:
150%;mso-list:l13 level1 lfo1;tab-stops:list 25.5pt'><![if !supportLists]><=
span
style=3D'mso-list:Ignore'>34.<span style=3D'font:7.0pt "Times New Roman"'>&=
nbsp;
</span></span><![endif]>3 tesadüfi değişkenin istatistikleri=
<span
style=3D'font-family:Symbol;mso-ascii-font-family:"Times New Roman";mso-han=
si-font-family:
"Times New Roman";mso-char-type:symbol;mso-symbol-font-family:Symbol'><span
style=3D'mso-char-type:symbol;mso-symbol-font-family:Symbol'>m</span></span=
><sub>C</sub>=3D75,
<span style=3D'font-family:Symbol;mso-ascii-font-family:"Times New Roman";
mso-hansi-font-family:"Times New Roman";mso-char-type:symbol;mso-symbol-fon=
t-family:
Symbol'><span style=3D'mso-char-type:symbol;mso-symbol-font-family:Symbol'>=
m</span></span><sub>I</sub>=3D20,
<span style=3D'font-family:Symbol;mso-ascii-font-family:"Times New Roman";
mso-hansi-font-family:"Times New Roman";mso-char-type:symbol;mso-symbol-fon=
t-family:
Symbol'><span style=3D'mso-char-type:symbol;mso-symbol-font-family:Symbol'>=
m</span></span><sub>G</sub>=3D5;
<span style=3D'font-family:Symbol;mso-ascii-font-family:"Times New Roman";
mso-hansi-font-family:"Times New Roman";mso-char-type:symbol;mso-symbol-fon=
t-family:
Symbol'><span style=3D'mso-char-type:symbol;mso-symbol-font-family:Symbol'>=
s</span></span><sub>C</sub>=3D10,
<span style=3D'font-family:Symbol;mso-ascii-font-family:"Times New Roman";
mso-hansi-font-family:"Times New Roman";mso-char-type:symbol;mso-symbol-fon=
t-family:
Symbol'><span style=3D'mso-char-type:symbol;mso-symbol-font-family:Symbol'>=
s</span></span><sub>I</sub>=3D5,
<span style=3D'font-family:Symbol;mso-ascii-font-family:"Times New Roman";
mso-hansi-font-family:"Times New Roman";mso-char-type:symbol;mso-symbol-fon=
t-family:
Symbol'><span style=3D'mso-char-type:symbol;mso-symbol-font-family:Symbol'>=
s</span></span><sub>G</sub>=3D1;
<span style=3D'font-family:Symbol;mso-ascii-font-family:"Times New Roman";
mso-hansi-font-family:"Times New Roman";mso-char-type:symbol;mso-symbol-fon=
t-family:
Symbol'><span style=3D'mso-char-type:symbol;mso-symbol-font-family:Symbol'>=
s</span></span><sub>CI</sub>=3D-50,
<span style=3D'font-family:Symbol;mso-ascii-font-family:"Times New Roman";
mso-hansi-font-family:"Times New Roman";mso-char-type:symbol;mso-symbol-fon=
t-family:
Symbol'><span style=3D'mso-char-type:symbol;mso-symbol-font-family:Symbol'>=
s</span></span><sub>CG</sub>=3D20,
<span style=3D'font-family:Symbol;mso-ascii-font-family:"Times New Roman";
mso-hansi-font-family:"Times New Roman";mso-char-type:symbol;mso-symbol-fon=
t-family:
Symbol'><span style=3D'mso-char-type:symbol;mso-symbol-font-family:Symbol'>=
s</span></span><sub>IG</sub>=3D
-10 ise Y=3DC+I+G ve S=3DI+G tesadüfi değişkenlerinin <span
class=3DSpellE>kovaryansını</span> hesaplayanız.</p>
<p class=3DMsoNormal style=3D'margin-left:25.5pt;text-indent:-25.5pt;line-h=
eight:
150%;mso-list:l13 level1 lfo1;tab-stops:list 25.5pt'><![if !supportLists]><=
span
style=3D'mso-list:Ignore'>35.<span style=3D'font:7.0pt "Times New Roman"'>&=
nbsp;
</span></span><![endif]>Bir X tesadüfi değişkeninin orijine
göre 1. ve 2. momentleri sırasıyla 4 ve 25’dir. Da =
7;ılımı
bilinmeyen Z=3DX(3-X) tesadüfi değişkeninin E(Z) ve E(Z<sup>=
2</sup>)
parametrelerini yaklaşık tahmin ediniz.</p>
<p class=3DMsoNormal style=3D'margin-left:25.5pt;text-indent:-25.5pt;line-h=
eight:
150%;mso-list:l13 level1 lfo1;tab-stops:list 25.5pt'><![if !supportLists]><=
span
style=3D'mso-list:Ignore'>36.<span style=3D'font:7.0pt "Times New Roman"'>&=
nbsp;
</span></span><![endif]>Z=3DX<sup>2</sup>Y tesadüfi değişken=
inin
beklenen değer ve varyansını aşağıdaki bilgil=
er
ışığında yaklaşık hesaplayınız=
. <sub><!--[if gte vml 1]><v:shape
id=3D"_x0000_i1035" type=3D"#_x0000_t75" style=3D'width:204.75pt;height:33=
.75pt'
o:ole=3D"">
<v:imagedata src=3D"matist_sorular_dosyalar/image027.wmz" o:title=3D""/>
</v:shape><![endif]--><![if !vml]><img width=3D273 height=3D45
src=3D"matist_sorular_dosyalar/image028.gif" v:shapes=3D"_x0000_i1035"><![e=
ndif]></sub><!--[if gte mso 9]><xml>
<o:OLEObject Type=3D"Embed" ProgID=3D"Equation.3" ShapeID=3D"_x0000_i1035"
DrawAspect=3D"Content" ObjectID=3D"_1198912102">
</o:OLEObject>
</xml><![endif]--></p>
<p class=3DMsoNormal style=3D'margin-left:25.5pt;text-indent:-25.5pt;line-h=
eight:
150%;mso-list:l13 level1 lfo1;tab-stops:list 25.5pt'><![if !supportLists]><=
span
style=3D'mso-list:Ignore'>37.<span style=3D'font:7.0pt "Times New Roman"'>&=
nbsp;
</span></span><![endif]>Ortalamaları sırasıyla 0.3 ve 1.2 ol=
an X
ve Y tesadüfi değişkenleri ile ilgili 2. dereceden ortalamaya
göre momentleri aşağıda verilmiştir. Bir Z=3D Y+e<=
sup>-X</sup>
tesadüfi değişkeninin beklenen değer ve
varyansını yaklaşık hesaplayınız. <span
class=3DSpellE>m<sub>x</sub></span><sub>,2</sub>=3D0.03, <span class=3DSpel=
lE>m<sub>y</sub></span><sub>,2</sub>=3D0</p>
<p class=3DMsoNormal style=3D'margin-left:25.5pt;text-indent:-25.5pt;line-h=
eight:
150%;mso-list:l13 level1 lfo1;tab-stops:list 25.5pt'><![if !supportLists]><=
span
style=3D'mso-list:Ignore'>38.<span style=3D'font:7.0pt "Times New Roman"'>&=
nbsp;
</span></span><![endif]>Ortalama ve varyansları bilinen ili tesadü=
;fi
değişkenin bir fonksiyonu olan Z=3D3X<sup>2</sup>Y/2 ve T=3DX<sup=
>2</sup>/2Y
değişkenleri için beklenen değer ve varyans
değerlerini yaklaşık hesaplayınız. <span
style=3D'font-family:Symbol;mso-ascii-font-family:"Times New Roman";mso-han=
si-font-family:
"Times New Roman";mso-char-type:symbol;mso-symbol-font-family:Symbol'><span
style=3D'mso-char-type:symbol;mso-symbol-font-family:Symbol'>m</span></span=
><sub>x</sub>=3D10,
<span style=3D'font-family:Symbol;mso-ascii-font-family:"Times New Roman";
mso-hansi-font-family:"Times New Roman";mso-char-type:symbol;mso-symbol-fon=
t-family:
Symbol'><span style=3D'mso-char-type:symbol;mso-symbol-font-family:Symbol'>=
m</span></span><sub>y</sub>=3D4,
<span style=3D'font-family:Symbol;mso-ascii-font-family:"Times New Roman";
mso-hansi-font-family:"Times New Roman";mso-char-type:symbol;mso-symbol-fon=
t-family:
Symbol'><span style=3D'mso-char-type:symbol;mso-symbol-font-family:Symbol'>=
s</span></span><sub>x</sub>=3D2,
<span style=3D'font-family:Symbol;mso-ascii-font-family:"Times New Roman";
mso-hansi-font-family:"Times New Roman";mso-char-type:symbol;mso-symbol-fon=
t-family:
Symbol'><span style=3D'mso-char-type:symbol;mso-symbol-font-family:Symbol'>=
s</span></span><sub>y</sub>=3D1.</p>
<p class=3DMsoNormal style=3D'margin-left:25.5pt;text-indent:-25.5pt;line-h=
eight:
150%;mso-list:l13 level1 lfo1;tab-stops:list 25.5pt'><![if !supportLists]><=
span
style=3D'mso-list:Ignore'>39.<span style=3D'font:7.0pt "Times New Roman"'>&=
nbsp;
</span></span><![endif]>X ve Y tesadüfi değişkenleri ile ilg=
ili
1. ve 2. dereceden orijine göre momentleri aşağıda
verilmiştir. Bir Z=3D e<sup>-(X+Y)</sup> tesadüfi
değişkeninin beklenen değer ve varyansını
yaklaşık hesaplayınız. <span class=3DSpellE>m<sub>x</su=
b></span><sub>,1</sub>=3D0.3,
<span class=3DSpellE>m<sub>x</sub></span><sub>,2</sub>=3D0.11, <span class=
=3DSpellE>m<sub>y</sub></span><sub>,1</sub>=3D1.2,
<span class=3DSpellE>m<sub>y</sub></span><sub>,2</sub>=3D1.5</p>
<p class=3DMsoNormal style=3D'margin-left:25.5pt;text-indent:-25.5pt;line-h=
eight:
150%;mso-list:l13 level1 lfo1;tab-stops:list 25.5pt'><![if !supportLists]><=
span
style=3D'mso-list:Ignore'>40.<span style=3D'font:7.0pt "Times New Roman"'>&=
nbsp;
</span></span><![endif]>X ve Y bağımsız tesadüfi
değişkenleri beklenen değer ve standart sapmaları E(X)=
=3D12,
S(X)=3D3, E(Y)=3D4, S(Y)=3D1 olarak bilinmektedir. Z=3DXY<sup>2</sup> olara=
k belirlenen
tesadüfi değişkenin beklenen değer ve standart
sapmasının yaklaşık değerlerini
hesaplayınız.</p>
<p class=3DMsoNormal style=3D'margin-left:25.5pt;text-indent:-25.5pt;line-h=
eight:
150%;mso-list:l13 level1 lfo1;tab-stops:list 25.5pt'><![if !supportLists]><=
span
style=3D'mso-list:Ignore'>41.<span style=3D'font:7.0pt "Times New Roman"'>&=
nbsp;
</span></span><![endif]>X ve Y bağımsız tesadüfi
değişkenleri beklenen değer ve standart sapmaları E(X)=
=3D12,
S(X)=3D3, E(Y)=3D4, S(Y)=3D1 olarak bilinmektedir. Z=3D20X<sup>2</sup>-3XY =
olarak
belirlenen tesadüfi değişkenin beklenen değer ve
varyansının yaklaşık değerlerini
hesaplayınız.</p>
<p class=3DMsoNormal style=3D'margin-left:25.5pt;text-indent:-25.5pt;line-h=
eight:
150%;mso-list:l13 level1 lfo1;tab-stops:list 25.5pt'><![if !supportLists]><=
span
style=3D'mso-list:Ignore'>42.<span style=3D'font:7.0pt "Times New Roman"'>&=
nbsp;
</span></span><![endif]>E[U(x)+V(y)]=3D E[U(x)]+E[V(y)] olduğunu kesik=
li
tesadüfi değişkenler için gösteriniz.</p>
<p class=3DMsoNormal style=3D'margin-left:25.5pt;text-indent:-25.5pt;line-h=
eight:
150%;mso-list:l13 level1 lfo1;tab-stops:list 25.5pt'><![if !supportLists]><=
span
style=3D'mso-list:Ignore'>43.<span style=3D'font:7.0pt "Times New Roman"'>&=
nbsp;
</span></span><![endif]>E[(X-E(X))(Y-E(Y))]=3DE(XY)-E(X)E(Y) olduğunu =
kesikli
tesadüfi değişkenler için gösteriniz.</p>
<p class=3DMsoNormal style=3D'margin-left:25.5pt;text-indent:-25.5pt;line-h=
eight:
150%;mso-list:l13 level1 lfo1;tab-stops:list 25.5pt'><![if !supportLists]><=
span
style=3D'mso-list:Ignore'>44.<span style=3D'font:7.0pt "Times New Roman"'>&=
nbsp;
</span></span><![endif]>V(Y)=3DE[V(Y/X)]+V[E(Y/X)] olduğunu göste=
riniz.</p>
<h1>ŞEKİLLER</h1>
<p class=3DMsoNormal style=3D'margin-left:25.5pt;text-indent:-25.5pt;line-h=
eight:
150%;mso-list:l13 level1 lfo1;tab-stops:list 25.5pt'><![if !supportLists]><=
span
style=3D'mso-list:Ignore'>45.<span style=3D'font:7.0pt "Times New Roman"'>&=
nbsp;
</span></span><![endif]>Aşağıda verilen frekans serisi
için olasılık fonksiyonunu ve dağılım
fonksiyonunu matematiksel olarak yazınız.</p>
<table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0
style=3D'margin-left:35.45pt;border-collapse:collapse;mso-yfti-tbllook:191;
mso-padding-alt:0cm 3.5pt 0cm 3.5pt'>
<tr>
<td width=3D106 valign=3Dtop style=3D'width:79.4pt;border-top:none;border=
-left:
none;border-bottom:solid black 1.5pt;border-right:solid black 1.5pt;
padding:0cm 3.5pt 0cm 3.5pt'>
<p class=3DMsoNormal style=3D'margin-top:6.0pt'><b>Çocuk Sayı=
sı<o:p></o:p></b></p>
</td>
<td width=3D89 valign=3Dtop style=3D'width:66.7pt;border:none;border-bott=
om:solid black 1.5pt;
mso-border-left-alt:solid black 1.5pt;padding:0cm 3.5pt 0cm 3.5pt'>
<p class=3DMsoNormal style=3D'margin-top:6.0pt'><b>Aile Sayısı<=
o:p></o:p></b></p>
</td>
</tr>
<tr>
<td width=3D106 valign=3Dtop style=3D'width:79.4pt;border:none;border-rig=
ht:solid black 1.5pt;
mso-border-top-alt:solid black 1.5pt;padding:0cm 3.5pt 0cm 3.5pt'>
<p class=3DMsoNormal style=3D'margin-top:6.0pt'><b>0<o:p></o:p></b></p>
</td>
<td width=3D89 valign=3Dtop style=3D'width:66.7pt;border:none;mso-border-=
top-alt:
solid black 1.5pt;mso-border-left-alt:solid black 1.5pt;padding:0cm 3.5pt=
0cm 3.5pt'>
<p class=3DMsoNormal style=3D'margin-top:6.0pt'>5</p>
</td>
</tr>
<tr>
<td width=3D106 valign=3Dtop style=3D'width:79.4pt;border:none;border-rig=
ht:solid black 1.5pt;
padding:0cm 3.5pt 0cm 3.5pt'>
<p class=3DMsoNormal style=3D'margin-top:6.0pt'><b>1<o:p></o:p></b></p>
</td>
<td width=3D89 valign=3Dtop style=3D'width:66.7pt;border:none;mso-border-=
left-alt:
solid black 1.5pt;padding:0cm 3.5pt 0cm 3.5pt'>
<p class=3DMsoNormal style=3D'margin-top:6.0pt'>10</p>
</td>
</tr>
<tr>
<td width=3D106 valign=3Dtop style=3D'width:79.4pt;border:none;border-rig=
ht:solid black 1.5pt;
padding:0cm 3.5pt 0cm 3.5pt'>
<p class=3DMsoNormal style=3D'margin-top:6.0pt'><b>2<o:p></o:p></b></p>
</td>
<td width=3D89 valign=3Dtop style=3D'width:66.7pt;border:none;mso-border-=
left-alt:
solid black 1.5pt;padding:0cm 3.5pt 0cm 3.5pt'>
<p class=3DMsoNormal style=3D'margin-top:6.0pt'>15</p>
</td>
</tr>
<tr>
<td width=3D106 valign=3Dtop style=3D'width:79.4pt;border:none;border-rig=
ht:solid black 1.5pt;
padding:0cm 3.5pt 0cm 3.5pt'>
<p class=3DMsoNormal style=3D'margin-top:6.0pt'><b>3<o:p></o:p></b></p>
</td>
<td width=3D89 valign=3Dtop style=3D'width:66.7pt;border:none;mso-border-=
left-alt:
solid black 1.5pt;padding:0cm 3.5pt 0cm 3.5pt'>
<p class=3DMsoNormal style=3D'margin-top:6.0pt'>20</p>
</td>
</tr>
<tr style=3D'mso-yfti-lastrow:yes'>
<td width=3D106 valign=3Dtop style=3D'width:79.4pt;border:none;border-rig=
ht:solid black 1.5pt;
padding:0cm 3.5pt 0cm 3.5pt'>
<p class=3DMsoNormal style=3D'margin-top:6.0pt'><b>4<o:p></o:p></b></p>
</td>
<td width=3D89 valign=3Dtop style=3D'width:66.7pt;border:none;mso-border-=
left-alt:
solid black 1.5pt;padding:0cm 3.5pt 0cm 3.5pt'>
<p class=3DMsoNormal style=3D'margin-top:6.0pt'>25</p>
</td>
</tr>
</table>
<p class=3DMsoNormal style=3D'margin-left:25.5pt;text-indent:-25.5pt;line-h=
eight:
150%;mso-list:l13 level1 lfo1;tab-stops:list 25.5pt'><![if !supportLists]><=
span
style=3D'mso-list:Ignore'>46.<span style=3D'font:7.0pt "Times New Roman"'>&=
nbsp;
</span></span><![endif]>Bir X tesadüfi değişkeni
aşağıda grafiği görüldüğü
şekilde dağılmıştır. Olasılık
yoğunluk fonksiyonunu parabole yaklaştırarak belirleyiniz.<b=
r>
<!--[if gte vml 1]><v:shape id=3D"_x0000_i1036" type=3D"#_x0000_t75" style=
=3D'width:3in;
height:2in' o:ole=3D"">
<v:imagedata src=3D"matist_sorular_dosyalar/image029.wmz" o:title=3D""/>
</v:shape><![endif]--><![if !vml]><img width=3D288 height=3D192
src=3D"matist_sorular_dosyalar/image030.gif" v:shapes=3D"_x0000_i1036"><![e=
ndif]><!--[if gte mso 9]><xml>
<o:OLEObject Type=3D"Embed" ProgID=3D"Word.Picture.8" ShapeID=3D"_x0000_i1=
036"
DrawAspect=3D"Content" ObjectID=3D"_1198912103">
</o:OLEObject>
</xml><![endif]--></p>
<p class=3DMsoNormal style=3D'margin-left:25.5pt;text-indent:-25.5pt;line-h=
eight:
150%;mso-list:l13 level1 lfo1;tab-stops:list 25.5pt'><![if !supportLists]><=
span
style=3D'mso-list:Ignore'>47.<span style=3D'font:7.0pt "Times New Roman"'>&=
nbsp;
</span></span><![endif]>Bir X tesadüfi değişkeninin [-3,3]
aralığında aşağıdaki şekilde
dağıldığı varsayılmaktadır.
Olasılık yoğunluk fonksiyonunu yazınız. Standart
sapmasını tahmin ediniz. <!--[if gte vml 1]><v:shape id=3D"_x0000=
_i1037"
type=3D"#_x0000_t75" style=3D'width:3in;height:2in' o:ole=3D"">
<v:imagedata src=3D"matist_sorular_dosyalar/image031.wmz" o:title=3D""/>
</v:shape><![endif]--><![if !vml]><img width=3D288 height=3D192
src=3D"matist_sorular_dosyalar/image032.gif" v:shapes=3D"_x0000_i1037"><![e=
ndif]><!--[if gte mso 9]><xml>
<o:OLEObject Type=3D"Embed" ProgID=3D"Word.Picture.8" ShapeID=3D"_x0000_i1=
037"
DrawAspect=3D"Content" ObjectID=3D"_1198912104">
</o:OLEObject>
</xml><![endif]--></p>
<p class=3DMsoNormal style=3D'margin-left:25.5pt;text-indent:-25.5pt;line-h=
eight:
150%;mso-list:l13 level1 lfo1;tab-stops:list 25.5pt'><![if !supportLists]><=
span
style=3D'mso-list:Ignore'>48.<span style=3D'font:7.0pt "Times New Roman"'>&=
nbsp;
</span></span><![endif]>(X,Y) iki boyutlu tesadüfi değişkeni
aşağıda görüldüğü şekilde S
bölgesinde düzgün (<span class=3DSpellE>uniform</span>)
dağılmıştır. Ortak olasılık yoğunluk
fonksiyonunu bularak k(x/y<x-1) koşullu olasılık fonksiyo=
nunu
yazınız.<br>
<span style=3D'mso-spacerun:yes'> </span><!--[if gte vml 1]><v:shape i=
d=3D"_x0000_i1038"
type=3D"#_x0000_t75" style=3D'width:3in;height:2in' o:ole=3D"">
<v:imagedata src=3D"matist_sorular_dosyalar/image033.wmz" o:title=3D""/>
</v:shape><![endif]--><![if !vml]><img width=3D288 height=3D192
src=3D"matist_sorular_dosyalar/image034.gif" v:shapes=3D"_x0000_i1038"><![e=
ndif]><!--[if gte mso 9]><xml>
<o:OLEObject Type=3D"Embed" ProgID=3D"Word.Picture.8" ShapeID=3D"_x0000_i1=
038"
DrawAspect=3D"Content" ObjectID=3D"_1198912105">
</o:OLEObject>
</xml><![endif]--></p>
<p class=3DMsoNormal style=3D'margin-left:25.5pt;text-indent:-25.5pt;line-h=
eight:
150%;mso-list:l13 level1 lfo1;tab-stops:list 25.5pt'><![if !supportLists]><=
span
style=3D'mso-list:Ignore'>49.<span style=3D'font:7.0pt "Times New Roman"'>&=
nbsp;
</span></span><![endif]>Bir X tesadüfi değişkeninin [-3,3]
aralığında aşağıdaki şekilde
dağıldığı varsayılmaktadır.
Olasılık yoğunluk fonksiyonunu yazınız. Medyan ve =
beklenen
değeri tahmin ederek asimetri durumunu yorumlayınız.<br>
<a name=3D"OLE_LINK2"><!--[if gte vml 1]><v:shape id=3D"_x0000_i1039" type=
=3D"#_x0000_t75"
style=3D'width:3in;height:2in' o:ole=3D"">
<v:imagedata src=3D"matist_sorular_dosyalar/image035.wmz" o:title=3D""/>
</v:shape><![endif]--><![if !vml]><img width=3D288 height=3D192
src=3D"matist_sorular_dosyalar/image036.gif" v:shapes=3D"_x0000_i1039"><![e=
ndif]><!--[if gte mso 9]><xml>
<o:OLEObject Type=3D"Embed" ProgID=3D"Word.Picture.8" ShapeID=3D"_x0000_i1=
039"
DrawAspect=3D"Content" ObjectID=3D"_1198912106">
</o:OLEObject>
</xml><![endif]--></a></p>
<p class=3DMsoNormal style=3D'margin-left:25.5pt;text-indent:-25.5pt;line-h=
eight:
150%;mso-list:l13 level1 lfo1;tab-stops:list 25.5pt'><![if !supportLists]><=
span
style=3D'mso-list:Ignore'>50.<span style=3D'font:7.0pt "Times New Roman"'>&=
nbsp;
</span></span><![endif]>Dağılımı tam olarak bilinmeyen =
-1<span
style=3D'font-family:Symbol;mso-ascii-font-family:"Times New Roman";mso-han=
si-font-family:
"Times New Roman";mso-char-type:symbol;mso-symbol-font-family:Symbol'><span
style=3D'mso-char-type:symbol;mso-symbol-font-family:Symbol'>£</span>=
</span>x<span
style=3D'font-family:Symbol;mso-ascii-font-family:"Times New Roman";mso-han=
si-font-family:
"Times New Roman";mso-char-type:symbol;mso-symbol-font-family:Symbol'><span
style=3D'mso-char-type:symbol;mso-symbol-font-family:Symbol'>£</span>=
</span>1
tanımlı X tesadüfi değişkeninin
aşağıdaki şekildeki olasılık yoğunluk
fonksiyonuna sahip olduğu gözlenmiştir. Parabole
yaklaştırarak varyansını tahmin ediniz.<br>
<!--[if gte vml 1]><v:shape id=3D"_x0000_i1040" type=3D"#_x0000_t75" style=
=3D'width:3in;
height:2in' o:ole=3D"">
<v:imagedata src=3D"matist_sorular_dosyalar/image037.wmz" o:title=3D""/>
</v:shape><![endif]--><![if !vml]><img width=3D288 height=3D192
src=3D"matist_sorular_dosyalar/image038.gif" v:shapes=3D"_x0000_i1040"><![e=
ndif]><!--[if gte mso 9]><xml>
<o:OLEObject Type=3D"Embed" ProgID=3D"Word.Picture.8" ShapeID=3D"_x0000_i1=
040"
DrawAspect=3D"Content" ObjectID=3D"_1198912107">
</o:OLEObject>
</xml><![endif]--></p>
<p class=3DMsoNormal style=3D'margin-left:25.5pt;text-indent:-25.5pt;line-h=
eight:
150%;mso-list:l13 level1 lfo1;tab-stops:list 25.5pt'><![if !supportLists]><=
span
style=3D'mso-list:Ignore'>51.<span style=3D'font:7.0pt "Times New Roman"'>&=
nbsp;
</span></span><![endif]>Dağılımı tam olarak bilinmeyen =
0<span
style=3D'font-family:Symbol;mso-ascii-font-family:"Times New Roman";mso-han=
si-font-family:
"Times New Roman";mso-char-type:symbol;mso-symbol-font-family:Symbol'><span
style=3D'mso-char-type:symbol;mso-symbol-font-family:Symbol'>£</span>=
</span>x<span
style=3D'font-family:Symbol;mso-ascii-font-family:"Times New Roman";mso-han=
si-font-family:
"Times New Roman";mso-char-type:symbol;mso-symbol-font-family:Symbol'><span
style=3D'mso-char-type:symbol;mso-symbol-font-family:Symbol'>£</span>=
</span>3<sup>1/2</sup>
tanımlı X tesadüfi değişkeninin
aşağıdaki şekildeki olasılık yoğunluk
fonksiyonuna sahip olduğu gözlenmiştir. Parabole
yaklaştırarak varyansını tahmin ediniz.<br>
<!--[if gte vml 1]><v:shape id=3D"_x0000_i1041" type=3D"#_x0000_t75" style=
=3D'width:299.25pt;
height:175.5pt' o:ole=3D"">
<v:imagedata src=3D"matist_sorular_dosyalar/image039.png" o:title=3D""/>
</v:shape><![endif]--><![if !vml]><img width=3D399 height=3D234
src=3D"matist_sorular_dosyalar/image040.jpg" v:shapes=3D"_x0000_i1041"><![e=
ndif]><!--[if gte mso 9]><xml>
<o:OLEObject Type=3D"Embed" ProgID=3D"PBrush" ShapeID=3D"_x0000_i1041"
DrawAspect=3D"Content" ObjectID=3D"_1198912108">
</o:OLEObject>
</xml><![endif]--></p>
<p class=3DMsoNormal style=3D'margin-left:25.5pt;text-indent:-25.5pt;line-h=
eight:
150%;mso-list:l13 level1 lfo1;tab-stops:list 25.5pt'><![if !supportLists]><=
span
style=3D'mso-list:Ignore'>52.<span style=3D'font:7.0pt "Times New Roman"'>&=
nbsp;
</span></span><![endif]>Bir X tesadüfi değişkeninin
dağılımı merkezi orijinde bulunan bir yarım daire
şeklindedir. Olasılık yoğunluk fonksiyonunu
yazınız.<br>
<br>
<sub><!--[if gte vml 1]><v:shape id=3D"_x0000_i1042" type=3D"#_x0000_t75" s=
tyle=3D'width:237.75pt;
height:36pt' o:ole=3D"">
<v:imagedata src=3D"matist_sorular_dosyalar/image041.wmz" o:title=3D""/>
</v:shape><![endif]--><![if !vml]><img width=3D317 height=3D48
src=3D"matist_sorular_dosyalar/image042.gif" v:shapes=3D"_x0000_i1042"><![e=
ndif]></sub><!--[if gte mso 9]><xml>
<o:OLEObject Type=3D"Embed" ProgID=3D"Equation.3" ShapeID=3D"_x0000_i1042"
DrawAspect=3D"Content" ObjectID=3D"_1198912109">
</o:OLEObject>
</xml><![endif]--></p>
<h1>DAĞILIMLARA İLİŞKİN ÖZEL SORULAR</h1>
<p class=3DMsoNormal style=3D'margin-left:25.5pt;text-indent:-25.5pt;line-h=
eight:
150%;mso-list:l13 level1 lfo1;tab-stops:list 25.5pt'><![if !supportLists]><=
span
style=3D'mso-list:Ignore'>53.<span style=3D'font:7.0pt "Times New Roman"'>&=
nbsp;
</span></span><![endif]>X tesadüfi değişkeninin karesinin
beklenen değerini ve P(X>2) olasılığını
hesaplayınız. <sub><!--[if gte vml 1]><v:shape id=3D"_x0000_i1043"
type=3D"#_x0000_t75" style=3D'width:189.75pt;height:60pt' o:ole=3D"">
<v:imagedata src=3D"matist_sorular_dosyalar/image043.wmz" o:title=3D""/>
</v:shape><![endif]--><![if !vml]><img width=3D253 height=3D80
src=3D"matist_sorular_dosyalar/image044.gif" v:shapes=3D"_x0000_i1043"><![e=
ndif]></sub><!--[if gte mso 9]><xml>
<o:OLEObject Type=3D"Embed" ProgID=3D"Equation.3" ShapeID=3D"_x0000_i1043"
DrawAspect=3D"Content" ObjectID=3D"_1198912110">
</o:OLEObject>
</xml><![endif]--></p>
<p class=3DMsoNormal style=3D'margin-left:25.5pt;text-indent:-25.5pt;line-h=
eight:
150%;mso-list:l13 level1 lfo1;tab-stops:list 25.5pt'><![if !supportLists]><=
span
style=3D'mso-list:Ignore'>54.<span style=3D'font:7.0pt "Times New Roman"'>&=
nbsp;
</span></span><![endif]><sub><!--[if gte vml 1]><v:shape id=3D"_x0000_i1064"
type=3D"#_x0000_t75" style=3D'width:222pt;height:54.75pt' o:ole=3D"">
<v:imagedata src=3D"matist_sorular_dosyalar/image045.wmz" o:title=3D""/>
</v:shape><![endif]--><![if !vml]><img width=3D296 height=3D73
src=3D"matist_sorular_dosyalar/image046.gif" v:shapes=3D"_x0000_i1064"><![e=
ndif]></sub><!--[if gte mso 9]><xml>
<o:OLEObject Type=3D"Embed" ProgID=3D"Equation.2" ShapeID=3D"_x0000_i1064"
DrawAspect=3D"Content" ObjectID=3D"_1198912111">
</o:OLEObject>
</xml><![endif]--><span style=3D'mso-spacerun:yes'> </span>olduğu=
nu
gösteriniz</p>
<p class=3DMsoNormal style=3D'margin-left:25.5pt;text-indent:-25.5pt;line-h=
eight:
150%;mso-list:l13 level1 lfo1;tab-stops:list 25.5pt'><![if !supportLists]><=
span
style=3D'mso-list:Ignore'>55.<span style=3D'font:7.0pt "Times New Roman"'>&=
nbsp;
</span></span><![endif]>X<sub>1</sub>, X<sub>2</sub>, ...., <span class=3DS=
pellE>X<sub>n</sub></span>
n tane p<sub>1</sub>,p<sub>2</sub>,...,<span class=3DSpellE>p<sub>n</sub></=
span>
parametreli bağımsız <span class=3DSpellE>Bernouilli</span>
değişkeni ise Y=3DX<sub>1</sub>+X<sub>2</sub>+...+<span class=3DS=
pellE>X<sub>n</sub></span>
için Y tesadüfi değişkeninin moment çıkar=
an
fonksiyonunu ve elde edilen teorik dağılımın
adını yazınız.</p>
<p class=3DMsoNormal style=3D'margin-left:25.5pt;text-indent:-25.5pt;line-h=
eight:
150%;mso-list:l13 level1 lfo1;tab-stops:list 25.5pt'><![if !supportLists]><=
span
style=3D'mso-list:Ignore'>56.<span style=3D'font:7.0pt "Times New Roman"'>&=
nbsp;
</span></span><![endif]>X<sub>1</sub>, X<sub>2</sub>, ...., <span class=3DS=
pellE>X<sub>n</sub></span>
n tane p<sub>1</sub>,p<sub>2</sub>,...,<span class=3DSpellE>p<sub>n</sub></=
span>
parametreli bağımsız <span class=3DSpellE>Bernouilli</span>
değişkeni ise Y=3D<span style=3D'font-family:Symbol;mso-ascii-fon=
t-family:
"Times New Roman";mso-hansi-font-family:"Times New Roman";mso-char-type:sym=
bol;
mso-symbol-font-family:Symbol'><span style=3D'mso-char-type:symbol;mso-symb=
ol-font-family:
Symbol'>b</span></span><sub>0</sub>+<span style=3D'font-family:Symbol;mso-a=
scii-font-family:
"Times New Roman";mso-hansi-font-family:"Times New Roman";mso-char-type:sym=
bol;
mso-symbol-font-family:Symbol'><span style=3D'mso-char-type:symbol;mso-symb=
ol-font-family:
Symbol'>b</span></span><sub>1</sub>X<sub>1</sub>+<span style=3D'font-family=
:Symbol;
mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:"Times New Ro=
man";
mso-char-type:symbol;mso-symbol-font-family:Symbol'><span style=3D'mso-char=
-type:
symbol;mso-symbol-font-family:Symbol'>b</span></span><sub>2</sub>X<sub>2</s=
ub>+...+<span
style=3D'font-family:Symbol;mso-ascii-font-family:"Times New Roman";mso-han=
si-font-family:
"Times New Roman";mso-char-type:symbol;mso-symbol-font-family:Symbol'><span
style=3D'mso-char-type:symbol;mso-symbol-font-family:Symbol'>b</span></span=
><span
class=3DSpellE><sub>n</sub>X<sub>n</sub></span> için Y tesadüfi
değişkeninin moment çıkaran fonksiyonunu
yazınız. (<span style=3D'font-family:Symbol;mso-ascii-font-family=
:"Times New Roman";
mso-hansi-font-family:"Times New Roman";mso-char-type:symbol;mso-symbol-fon=
t-family:
Symbol'><span style=3D'mso-char-type:symbol;mso-symbol-font-family:Symbol'>=
a</span></span>,
<span style=3D'font-family:Symbol;mso-ascii-font-family:"Times New Roman";
mso-hansi-font-family:"Times New Roman";mso-char-type:symbol;mso-symbol-fon=
t-family:
Symbol'><span style=3D'mso-char-type:symbol;mso-symbol-font-family:Symbol'>=
b</span></span><sub>k</sub>
0’dan farklı sabit sayılardır)</p>
<p class=3DMsoNormal style=3D'margin-left:25.5pt;text-indent:-25.5pt;line-h=
eight:
150%;mso-list:l13 level1 lfo1;tab-stops:list 25.5pt'><![if !supportLists]><=
span
style=3D'mso-list:Ignore'>57.<span style=3D'font:7.0pt "Times New Roman"'>&=
nbsp;
</span></span><![endif]>İçinde çok sayıda
kırmızı ve beyaz topların bulunduğu bir torbadan
seçilen bir topun kırmızı olması (<span
class=3DSpellE>X<sub>i</sub></span>) olasılığı %30̵=
7;dur.
Arka arkaya tekrarlanan bağımsız denemeler ile torbadan 10 t=
op
tesadüfi olarak seçilecektir. Kırmızı toplar =
5;n
sayısına ilişkin (Y) moment çıkartan fonksiyonu
belirleyiniz.</p>
<p class=3DMsoNormal style=3D'margin-left:25.5pt;text-indent:-25.5pt;line-h=
eight:
150%;mso-list:l13 level1 lfo1;tab-stops:list 25.5pt'><![if !supportLists]><=
span
style=3D'mso-list:Ignore'>58.<span style=3D'font:7.0pt "Times New Roman"'>&=
nbsp;
</span></span><![endif]>B olayı n adet hilesiz madeni paranın
atılması deneyidir ve “<span class=3DSpellE>B<sub>i</sub></=
span>”
oluşan muhtemel sonuçlardır. Alt olayların hiç=
biri
boş olmayıp aynı olasılığa [P(<span class=3DS=
pellE>Bi</span>)=3Db]
sahiptir. Ayrıca A, <span class=3DSpellE>B’nin</span> bir ba#=
1;ka
alt olayıdır. Bir “<span class=3DSpellE>B<sub>i</sub></span=
>”
koşulunun gerçekleştiği bilindiğine göre, A
olayının meydana gelmesi olasılığının P(=
A/<span
class=3DSpellE>B<sub>i</sub></span>)=3D<span style=3D'font-family:Symbol;ms=
o-ascii-font-family:
"Times New Roman";mso-hansi-font-family:"Times New Roman";mso-char-type:sym=
bol;
mso-symbol-font-family:Symbol'><span style=3D'mso-char-type:symbol;mso-symb=
ol-font-family:
Symbol'>l</span></span><span style=3D'font-family:Symbol;mso-ascii-font-fam=
ily:
"Times New Roman";mso-hansi-font-family:"Times New Roman";mso-char-type:sym=
bol;
mso-symbol-font-family:Symbol'><span style=3D'mso-char-type:symbol;mso-symb=
ol-font-family:
Symbol'>×</span></span>b (<span style=3D'font-family:Symbol;mso-ascii=
-font-family:
"Times New Roman";mso-hansi-font-family:"Times New Roman";mso-char-type:sym=
bol;
mso-symbol-font-family:Symbol'><span style=3D'mso-char-type:symbol;mso-symb=
ol-font-family:
Symbol'>l</span></span>:sabit) olduğu bilinmektedir. (b)’<span
class=3DSpellE>nin</span> değeri kaçtır ve P(<span class=
=3DSpellE>B<sub>i</sub></span>/A)=3D?</p>
<p class=3DMsoNormal style=3D'margin-left:25.5pt;text-indent:-25.5pt;line-h=
eight:
150%;mso-list:l13 level1 lfo1;tab-stops:list 25.5pt'><![if !supportLists]><=
span
style=3D'mso-list:Ignore'>59.<span style=3D'font:7.0pt "Times New Roman"'>&=
nbsp;
</span></span><![endif]>X ve Y bağımsız tesadüfi
değişkenleri 4<span style=3D'font-family:Symbol;mso-ascii-font-fa=
mily:
"Times New Roman";mso-hansi-font-family:"Times New Roman";mso-char-type:sym=
bol;
mso-symbol-font-family:Symbol'><span style=3D'mso-char-type:symbol;mso-symb=
ol-font-family:
Symbol'>£</span></span>X<span style=3D'font-family:Symbol;mso-ascii-f=
ont-family:
"Times New Roman";mso-hansi-font-family:"Times New Roman";mso-char-type:sym=
bol;
mso-symbol-font-family:Symbol'><span style=3D'mso-char-type:symbol;mso-symb=
ol-font-family:
Symbol'>£</span></span>9, 0<span style=3D'font-family:Symbol;mso-asci=
i-font-family:
"Times New Roman";mso-hansi-font-family:"Times New Roman";mso-char-type:sym=
bol;
mso-symbol-font-family:Symbol'><span style=3D'mso-char-type:symbol;mso-symb=
ol-font-family:
Symbol'>£</span></span>Y<span style=3D'font-family:Symbol;mso-ascii-f=
ont-family:
"Times New Roman";mso-hansi-font-family:"Times New Roman";mso-char-type:sym=
bol;
mso-symbol-font-family:Symbol'><span style=3D'mso-char-type:symbol;mso-symb=
ol-font-family:
Symbol'>£</span></span>1 aralığında <span class=3DSpel=
lE>üniform</span>
(düzgün) dağılmışlardır. U=3DX2/Y ise <s=
pan
class=3DSpellE>u’nun</span> olasılık yoğunluk fonksiyo=
nunu
yazınız.</p>
<p class=3DMsoNormal style=3D'margin-left:25.5pt;text-indent:-25.5pt;line-h=
eight:
150%;mso-list:l13 level1 lfo1;tab-stops:list 25.5pt'><![if !supportLists]><=
span
style=3D'mso-list:Ignore'>60.<span style=3D'font:7.0pt "Times New Roman"'>&=
nbsp;
</span></span><![endif]>Sürekli <span class=3DSpellE>uniform</span>
(düzgün) dağılmış X ve Y tesadüfi
değişkenleri (0, 1) aralığında
tanımlanmışlardır. U=3DY/X iki boyutlu tesadüfi
değişkeninin olasılık yoğunluk ve
dağılım fonksiyonlarını yazınız.</p>
<p class=3DMsoNormal style=3D'margin-left:25.5pt;text-indent:-25.5pt;line-h=
eight:
150%;mso-list:l13 level1 lfo1;tab-stops:list 25.5pt'><![if !supportLists]><=
span
style=3D'mso-list:Ignore'>61.<span style=3D'font:7.0pt "Times New Roman"'>&=
nbsp;
</span></span><![endif]>Atatürk havaalanına, “THY”
uçakları ortalama 30 dakikada bir inerken “<span
class=3DSpellE>Swissair</span>” uçakları yaklaşı=
;k 2
saatte bir inmektedirler. Şirketlerin inişleri arasındaki
süreler sırasıyla X ve Y tesadüfi değişkenleri
ise Z=3DXY tesadüfi değişkeninin ortalamaya göre moment
çıkaran fonksiyonunu yazınız.</p>
<p class=3DMsoNormal style=3D'margin-left:25.5pt;text-indent:-25.5pt;line-h=
eight:
150%;mso-list:l13 level1 lfo1;tab-stops:list 25.5pt'><![if !supportLists]><=
span
style=3D'mso-list:Ignore'>62.<span style=3D'font:7.0pt "Times New Roman"'>&=
nbsp;
</span></span><![endif]>Atatürk havaalanına, “THY”
uçakları ortalama 60 dakikada bir inerken “<span
class=3DSpellE>Swissair</span>” uçakları yaklaşı=
;k 2
saatte bir ve <span class=3DSpellE>Lufthansa</span> uçakları 90
dakikada bir inmektedirler. Şirketlerin inişleri arasındaki
süreler sırasıyla X, Y ve Z tesadüfi değişken=
leri
ise L=3DX+Y+Z tesadüfi değişkeninin ortalamaya göre mom=
ent
çıkaran fonksiyonunu yazınız.</p>
<p class=3DMsoNormal style=3D'margin-left:25.5pt;text-indent:-25.5pt;line-h=
eight:
150%;mso-list:l13 level1 lfo1;tab-stops:list 25.5pt'><![if !supportLists]><=
span
style=3D'mso-list:Ignore'>63.<span style=3D'font:7.0pt "Times New Roman"'>&=
nbsp;
</span></span><![endif]>Bir <span class=3DSpellE>fast</span>-<span class=3D=
SpellE>food</span>
<span class=3DSpellE>restorantına</span> giren müşteriler
arasındaki ortalama süre 2 dakika olarak belirlenmiştir.
Yarım saatlik <span class=3DSpellE>period</span> içerisinde
müşteri girişleri arasındaki süre olan X tesad&uum=
l;fi
değişkeninin olasılık yoğunluk fonksiyonunu
yazınız.</p>
<p class=3DMsoNormal style=3D'margin-left:25.5pt;text-indent:-25.5pt;line-h=
eight:
150%;mso-list:l13 level1 lfo1;tab-stops:list 25.5pt'><![if !supportLists]><=
span
style=3D'mso-list:Ignore'>64.<span style=3D'font:7.0pt "Times New Roman"'>&=
nbsp;
</span></span><![endif]>Saat 13:00-17:00 arasında yarım saat
içinde bir limana ortalama 6 gemi girmektedir. Aynı zaman dilim=
inde
bir saat içinde bu limana giren gemilerin sayısı X
tesadüfi değişkeni iken olasılık fonksiyonunu
yazınız.</p>
<p class=3DMsoCaption>MOMENT ÇIKARAN FONKSİYON VE
ÖZELLİKLERİ</p>
<p class=3DMsoNormal style=3D'margin-left:25.5pt;text-indent:-25.5pt;line-h=
eight:
150%;mso-list:l13 level1 lfo1;tab-stops:list 25.5pt'><![if !supportLists]><=
span
style=3D'mso-list:Ignore'>65.<span style=3D'font:7.0pt "Times New Roman"'>&=
nbsp;
</span></span><![endif]><sub><!--[if gte vml 1]><v:shape id=3D"_x0000_i1044"
type=3D"#_x0000_t75" style=3D'width:110.25pt;height:38.25pt' o:ole=3D"">
<v:imagedata src=3D"matist_sorular_dosyalar/image047.wmz" o:title=3D""/>
</v:shape><![endif]--><![if !vml]><img width=3D147 height=3D51
src=3D"matist_sorular_dosyalar/image048.gif" v:shapes=3D"_x0000_i1044"><![e=
ndif]></sub><!--[if gte mso 9]><xml>
<o:OLEObject Type=3D"Embed" ProgID=3D"Equation.3" ShapeID=3D"_x0000_i1044"
DrawAspect=3D"Content" ObjectID=3D"_1198912112">
</o:OLEObject>
</xml><![endif]--><span style=3D'mso-spacerun:yes'> </span>verilen
olasılık yoğunluk fonksiyonunun ortalamaya göre moment
çıkaran fonksiyonunu bulunuz. Serinin simetrilik durumunu
inceleyiniz.</p>
<p class=3DMsoNormal style=3D'margin-left:25.5pt;text-indent:-25.5pt;line-h=
eight:
150%;mso-list:l13 level1 lfo1;tab-stops:list 25.5pt'><![if !supportLists]><=
span
style=3D'mso-list:Ignore'>66.<span style=3D'font:7.0pt "Times New Roman"'>&=
nbsp;
</span></span><![endif]>X tesadüfi değişkeninin orijine
göre moment çıkartan fonksiyonunu bulunuz. <sub><!--[if gt=
e vml 1]><v:shape
id=3D"_x0000_i1045" type=3D"#_x0000_t75" style=3D'width:96pt;height:30.75p=
t' o:ole=3D"">
<v:imagedata src=3D"matist_sorular_dosyalar/image049.wmz" o:title=3D""/>
</v:shape><![endif]--><![if !vml]><img width=3D128 height=3D41
src=3D"matist_sorular_dosyalar/image050.gif" v:shapes=3D"_x0000_i1045"><![e=
ndif]></sub><!--[if gte mso 9]><xml>
<o:OLEObject Type=3D"Embed" ProgID=3D"Equation.3" ShapeID=3D"_x0000_i1045"
DrawAspect=3D"Content" ObjectID=3D"_1198912113">
</o:OLEObject>
</xml><![endif]--></p>
<p class=3DMsoNormal style=3D'margin-left:25.5pt;text-indent:-25.5pt;line-h=
eight:
150%;mso-list:l13 level1 lfo1;tab-stops:list 25.5pt'><![if !supportLists]><=
span
style=3D'mso-list:Ignore'>67.<span style=3D'font:7.0pt "Times New Roman"'>&=
nbsp;
</span></span><![endif]>f(x)=3D(1/2)x, x=3D1,2,..... olasılık
yoğunluk fonksiyonuna sahip X tesadüfi değişkeninin mom=
ent
çıkaran fonksiyonunu belirleyerek ikinci dereceden orijine
göre momentini belirleyiniz.</p>
<p class=3DMsoNormal style=3D'margin-left:25.5pt;text-indent:-25.5pt;line-h=
eight:
150%;mso-list:l13 level1 lfo1;tab-stops:list 25.5pt'><![if !supportLists]><=
span
style=3D'mso-list:Ignore'>68.<span style=3D'font:7.0pt "Times New Roman"'>&=
nbsp;
</span></span><![endif]>Olasılık yoğunluk fonksiyonu verilen=
X
tesadüfi değişkeninin “moment çıkaran
fonksiyon”unu belirleyerek, ortalamaya göre moment
çıkaran fonksiyonunu gösteriniz ve basıklık duru=
munu
inceleyiniz.<br>
<sub><!--[if gte vml 1]><v:shape id=3D"_x0000_i1046" type=3D"#_x0000_t75" s=
tyle=3D'width:200.25pt;
height:38.25pt' o:ole=3D"">
<v:imagedata src=3D"matist_sorular_dosyalar/image051.wmz" o:title=3D""/>
</v:shape><![endif]--><![if !vml]><img width=3D267 height=3D51
src=3D"matist_sorular_dosyalar/image052.gif" v:shapes=3D"_x0000_i1046"><![e=
ndif]></sub><!--[if gte mso 9]><xml>
<o:OLEObject Type=3D"Embed" ProgID=3D"Equation.3" ShapeID=3D"_x0000_i1046"
DrawAspect=3D"Content" ObjectID=3D"_1198912114">
</o:OLEObject>
</xml><![endif]--></p>
<p class=3DMsoNormal style=3D'margin-left:25.5pt;text-indent:-25.5pt;line-h=
eight:
150%;mso-list:l13 level1 lfo1;tab-stops:list 25.5pt'><![if !supportLists]><=
span
style=3D'mso-list:Ignore'>69.<span style=3D'font:7.0pt "Times New Roman"'>&=
nbsp;
</span></span><![endif]>Bir X tesadüfi değişkeninin moment
çıkartan fonksiyonu M<sub>X</sub>(t)=3D<span style=3D'font-size=
:14.0pt;
line-height:150%'>e</span><sup><span style=3D'font-size:16.0pt;line-height:=
150%'>3t+8t²</span></sup>
ise Z=3D¼(X-3) tesadüfi değişkeninin moment
çıkartan fonksiyonunu yazın ve beklenen değerini
belirleyin.</p>
<p class=3DMsoNormal style=3D'margin-left:25.5pt;text-indent:-25.5pt;line-h=
eight:
150%;mso-list:l13 level1 lfo1;tab-stops:list 25.5pt'><![if !supportLists]><=
span
style=3D'mso-list:Ignore'>70.<span style=3D'font:7.0pt "Times New Roman"'>&=
nbsp;
</span></span><![endif]>Bir T tesadüfi değişkeni moment
çıkaran fonksiyonları bilinen bağımsız X, Y
tesadüfi değişkenlerinden T=3D2X+Y/3-6 şeklinde meydana
gelmektedir. T tesadüfi değişkeninin beklenen değerini
momentler yardımıyla bulunuz. <span class=3DSpellE>M<sub>x</sub><=
/span>(t)=3De<sup>t/6</sup>,<span
class=3DSpellE>M<sub>y</sub></span>(t)=3D1/e<sup>2t</sup>.</p>
<p class=3DMsoNormal style=3D'margin-left:25.5pt;text-indent:-25.5pt;line-h=
eight:
150%;mso-list:l13 level1 lfo1;tab-stops:list 25.5pt'><![if !supportLists]><=
span
style=3D'mso-list:Ignore'>71.<span style=3D'font:7.0pt "Times New Roman"'>&=
nbsp;
</span></span><![endif]>Bir T tesadüfi değişkeni moment
çıkaran fonksiyonları bilinen X, Y, Z tesadüfi
değişkenlerinden T=3D2X+3Y-6Z şeklinde meydana gelmektedir. T
tesadüfi değişkeninin beklenen değerini momentler
yardımıyla bulunuz. <span class=3DSpellE>M<sub>x</sub></span>(t)=
=3De<sup>t/6</sup>,<span
class=3DSpellE>M<sub>y</sub></span>(t)=3D1/e<sup>2t</sup>,<span class=3DSpe=
llE>M<sub>z</sub></span>(t)=3De<sup>t</sup>.</p>
<p class=3DMsoNormal style=3D'margin-left:25.5pt;text-indent:-25.5pt;line-h=
eight:
150%;mso-list:l13 level1 lfo1;tab-stops:list 25.5pt'><![if !supportLists]><=
span
style=3D'mso-list:Ignore'>72.<span style=3D'font:7.0pt "Times New Roman"'>&=
nbsp;
</span></span><![endif]>X, Y, Z tesadüfi değişkenlerinin bir
fonksiyonu olan T=3D2+4X-Z-5Y tesadüfi değişkeninin ortalama=
ya
göre moment çıkaran fonksiyonunu yazınız.<br>
<sub><!--[if gte vml 1]><v:shape id=3D"_x0000_i1047" type=3D"#_x0000_t75" s=
tyle=3D'width:279.75pt;
height:35.25pt' o:ole=3D"">
<v:imagedata src=3D"matist_sorular_dosyalar/image053.wmz" o:title=3D""/>
</v:shape><![endif]--><![if !vml]><img width=3D373 height=3D47
src=3D"matist_sorular_dosyalar/image054.gif" v:shapes=3D"_x0000_i1047"><![e=
ndif]></sub><!--[if gte mso 9]><xml>
<o:OLEObject Type=3D"Embed" ProgID=3D"Equation.3" ShapeID=3D"_x0000_i1047"
DrawAspect=3D"Content" ObjectID=3D"_1198912115">
</o:OLEObject>
</xml><![endif]--></p>
<p class=3DMsoNormal style=3D'margin-left:25.5pt;text-indent:-25.5pt;line-h=
eight:
150%;mso-list:l13 level1 lfo1;tab-stops:list 25.5pt'><![if !supportLists]><=
span
style=3D'mso-list:Ignore'>73.<span style=3D'font:7.0pt "Times New Roman"'>&=
nbsp;
</span></span><![endif]>Bir T tesadüfi değişkeni moment
çıkaran fonksiyonları bilinen X ve Y tesadüfi
değişkenlerinden T=3D(X-<span style=3D'font-family:Symbol;mso-asc=
ii-font-family:
"Times New Roman";mso-hansi-font-family:"Times New Roman";mso-char-type:sym=
bol;
mso-symbol-font-family:Symbol'><span style=3D'mso-char-type:symbol;mso-symb=
ol-font-family:
Symbol'>m</span></span><sub>x</sub>)(Y-<span style=3D'font-family:Symbol;
mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:"Times New Ro=
man";
mso-char-type:symbol;mso-symbol-font-family:Symbol'><span style=3D'mso-char=
-type:
symbol;mso-symbol-font-family:Symbol'>m</span></span><sub>y</sub>)/<span
class=3DSpellE>cov</span>(X,Y), <span class=3DSpellE>cov</span>(X,Y)=3D<span
class=3DSpellE>e<sup>xy</sup></span> şeklinde meydana gelmektedir. T
tesadüfi değişkeninin beklenen değerini momentler
yardımıyla bulunuz. <span class=3DSpellE>M<sub>x</sub></span>(t)=
=3D<span
class=3DSpellE>M<sub>y</sub></span>(t)=3De<sup>t</sup>.</p>
<p class=3DMsoNormal style=3D'margin-left:25.5pt;text-indent:-25.5pt;line-h=
eight:
150%;mso-list:l13 level1 lfo1;tab-stops:list 25.5pt'><![if !supportLists]><=
span
style=3D'mso-list:Ignore'>74.<span style=3D'font:7.0pt "Times New Roman"'>&=
nbsp;
</span></span><![endif]>Bir X~N(<span style=3D'font-family:Symbol;mso-ascii=
-font-family:
"Times New Roman";mso-hansi-font-family:"Times New Roman";mso-char-type:sym=
bol;
mso-symbol-font-family:Symbol'><span style=3D'mso-char-type:symbol;mso-symb=
ol-font-family:
Symbol'>m</span></span>, <span style=3D'font-family:Symbol;mso-ascii-font-f=
amily:
"Times New Roman";mso-hansi-font-family:"Times New Roman";mso-char-type:sym=
bol;
mso-symbol-font-family:Symbol'><span style=3D'mso-char-type:symbol;mso-symb=
ol-font-family:
Symbol'>s</span></span><sup>2</sup>) tesadüfi değişkeninin
moment çıkartan fonksiyonu <sub><!--[if gte vml 1]><v:shape id=
=3D"_x0000_i1048"
type=3D"#_x0000_t75" style=3D'width:93pt;height:36pt' o:ole=3D"">
<v:imagedata src=3D"matist_sorular_dosyalar/image055.wmz" o:title=3D""/>
</v:shape><![endif]--><![if !vml]><img width=3D124 height=3D48
src=3D"matist_sorular_dosyalar/image056.gif" v:shapes=3D"_x0000_i1048"><![e=
ndif]></sub><!--[if gte mso 9]><xml>
<o:OLEObject Type=3D"Embed" ProgID=3D"Equation.3" ShapeID=3D"_x0000_i1048"
DrawAspect=3D"Content" ObjectID=3D"_1198912116">
</o:OLEObject>
</xml><![endif]--><span style=3D'mso-spacerun:yes'> </span>iken X
değerlerini standartlaştırarak elde edilen Z tesadüfi
değişkeninin (Z: <span class=3DSpellE>x’in</span> ortalama =
ile
farkının standart sapma cinsinden ifadesidir) orijine göre
moment çıkartan fonksiyonunu yazınız. İkinci
dereceden momenti bulacağınız denklemi yazınız.</p>
<h1>DAĞILIM FONKSİYONU TEKNİĞİ</h1>
<p class=3DMsoNormal style=3D'margin-left:25.5pt;text-indent:-25.5pt;line-h=
eight:
150%;mso-list:l13 level1 lfo1;tab-stops:list 25.5pt'><![if !supportLists]><=
span
style=3D'mso-list:Ignore'>75.<span style=3D'font:7.0pt "Times New Roman"'>&=
nbsp;
</span></span><![endif]>Aşağıda verilen X tesadüfi
değişkeninin olasılık fonksiyonundan yararlanarak Y=3DU=
(X)=3D2X
tesadüfi değişkeninin dağılım fonksiyonunu
monoton artan/azalan tekniğini kullanarak belirleyiniz. <sub><!--[if g=
te vml 1]><v:shape
id=3D"_x0000_i1049" type=3D"#_x0000_t75" style=3D'width:159.75pt;height:45=
.75pt'
o:ole=3D"">
<v:imagedata src=3D"matist_sorular_dosyalar/image057.wmz" o:title=3D""/>
</v:shape><![endif]--><![if !vml]><img width=3D213 height=3D61
src=3D"matist_sorular_dosyalar/image058.gif" v:shapes=3D"_x0000_i1049"><![e=
ndif]></sub><!--[if gte mso 9]><xml>
<o:OLEObject Type=3D"Embed" ProgID=3D"Equation.3" ShapeID=3D"_x0000_i1049"
DrawAspect=3D"Content" ObjectID=3D"_1198912117">
</o:OLEObject>
</xml><![endif]--></p>
<p class=3DMsoNormal style=3D'margin-left:25.5pt;text-indent:-25.5pt;line-h=
eight:
150%;mso-list:l13 level1 lfo1;tab-stops:list 25.5pt'><![if !supportLists]><=
span
style=3D'mso-list:Ignore'>76.<span style=3D'font:7.0pt "Times New Roman"'>&=
nbsp;
</span></span><![endif]>Aşağıda verilen X tesadüfi
değişkeninin olasılık yoğunluk fonksiyonundan
yararlanarak Y=3DU(X)=3D1-2X tesadüfi değişkeninin
dağılım fonksiyonunu monoton artan/azalan tekniğini kul=
lanarak
belirleyiniz.<br>
<sub><!--[if gte vml 1]><v:shape id=3D"_x0000_i1050" type=3D"#_x0000_t75" s=
tyle=3D'width:162.75pt;
height:60pt' o:ole=3D"">
<v:imagedata src=3D"matist_sorular_dosyalar/image059.wmz" o:title=3D""/>
</v:shape><![endif]--><![if !vml]><img width=3D217 height=3D80
src=3D"matist_sorular_dosyalar/image060.gif" v:shapes=3D"_x0000_i1050"><![e=
ndif]></sub><!--[if gte mso 9]><xml>
<o:OLEObject Type=3D"Embed" ProgID=3D"Equation.3" ShapeID=3D"_x0000_i1050"
DrawAspect=3D"Content" ObjectID=3D"_1198912118">
</o:OLEObject>
</xml><![endif]--></p>
<p class=3DMsoNormal style=3D'margin-left:25.5pt;text-indent:-25.5pt;line-h=
eight:
150%;mso-list:l13 level1 lfo1;tab-stops:list 25.5pt'><![if !supportLists]><=
span
style=3D'mso-list:Ignore'>77.<span style=3D'font:7.0pt "Times New Roman"'>&=
nbsp;
</span></span><![endif]>Aşağıda verilen X tesadüfi
değişkeninin olasılık yoğunluk fonksiyonundan
yararlanarak Y=3DU(X)=3Dtan(X) tesadüfi değişkeninin
dağılım fonksiyonunu monoton artan/azalan tekniğini kul=
lanarak
belirleyiniz.<br>
<sub><!--[if gte vml 1]><v:shape id=3D"_x0000_i1051" type=3D"#_x0000_t75" s=
tyle=3D'width:138.75pt;
height:45.75pt' o:ole=3D"">
<v:imagedata src=3D"matist_sorular_dosyalar/image061.wmz" o:title=3D""/>
</v:shape><![endif]--><![if !vml]><img width=3D185 height=3D61
src=3D"matist_sorular_dosyalar/image062.gif" v:shapes=3D"_x0000_i1051"><![e=
ndif]></sub><!--[if gte mso 9]><xml>
<o:OLEObject Type=3D"Embed" ProgID=3D"Equation.3" ShapeID=3D"_x0000_i1051"
DrawAspect=3D"Content" ObjectID=3D"_1198912119">
</o:OLEObject>
</xml><![endif]--></p>
<p class=3DMsoNormal style=3D'margin-left:25.5pt;text-indent:-25.5pt;line-h=
eight:
150%;mso-list:l13 level1 lfo1;tab-stops:list 25.5pt'><![if !supportLists]><=
span
style=3D'mso-list:Ignore'>78.<span style=3D'font:7.0pt "Times New Roman"'>&=
nbsp;
</span></span><![endif]>Aşağıda verilen X tesadüfi
değişkeninin olasılık fonksiyonundan yararlanarak Y=3DU=
(X)
tesadüfi değişkeninin dağılım fonksiyonunu
monoton artan/azalan tekniğini kullanarak belirleyiniz. <sub><!--[if g=
te vml 1]><v:shape
id=3D"_x0000_i1052" type=3D"#_x0000_t75" style=3D'width:254.25pt;height:42=
.75pt'
o:ole=3D"">
<v:imagedata src=3D"matist_sorular_dosyalar/image063.wmz" o:title=3D""/>
</v:shape><![endif]--><![if !vml]><img width=3D339 height=3D57
src=3D"matist_sorular_dosyalar/image064.gif" v:shapes=3D"_x0000_i1052"><![e=
ndif]></sub><!--[if gte mso 9]><xml>
<o:OLEObject Type=3D"Embed" ProgID=3D"Equation.3" ShapeID=3D"_x0000_i1052"
DrawAspect=3D"Content" ObjectID=3D"_1198912120">
</o:OLEObject>
</xml><![endif]--></p>
<h1>DEĞİŞKEN DEĞİŞTİRME
TEKNİĞİ</h1>
<p class=3DMsoNormal style=3D'margin-left:25.5pt;text-indent:-25.5pt;line-h=
eight:
150%;mso-list:l13 level1 lfo1;tab-stops:list 25.5pt'><![if !supportLists]><=
span
style=3D'mso-list:Ignore'>79.<span style=3D'font:7.0pt "Times New Roman"'>&=
nbsp;
</span></span><![endif]>X tesadüfi değişkeninin olasıl&=
#305;k
yoğunluk fonksiyonu aşağıda verilmiştir. Y=3D1-X
fonksiyonunun olasılık yoğunluk fonksiyonunu değiş=
ken
değiştirme tekniği ile bulunuz.<br>
<sub><!--[if gte vml 1]><v:shape id=3D"_x0000_i1053" type=3D"#_x0000_t75" s=
tyle=3D'width:141.75pt;
height:36pt' o:ole=3D"">
<v:imagedata src=3D"matist_sorular_dosyalar/image065.wmz" o:title=3D""/>
</v:shape><![endif]--><![if !vml]><img width=3D189 height=3D48
src=3D"matist_sorular_dosyalar/image066.gif" v:shapes=3D"_x0000_i1053"><![e=
ndif]></sub><!--[if gte mso 9]><xml>
<o:OLEObject Type=3D"Embed" ProgID=3D"Equation.3" ShapeID=3D"_x0000_i1053"
DrawAspect=3D"Content" ObjectID=3D"_1198912121">
</o:OLEObject>
</xml><![endif]--></p>
<p class=3DMsoNormal style=3D'margin-left:25.5pt;text-indent:-25.5pt;line-h=
eight:
150%;mso-list:l13 level1 lfo1;tab-stops:list 25.5pt'><![if !supportLists]><=
span
style=3D'mso-list:Ignore'>80.<span style=3D'font:7.0pt "Times New Roman"'>&=
nbsp;
</span></span><![endif]>X tesadüfi değişkeninin
olasılık yoğunluk fonksiyonu aşağıda
verilmiştir. Y=3D3X<sup>2</sup> fonksiyonunun olasılık
yoğunluk fonksiyonunu değişken değiştirme tekni=
87;i
ile bulunuz.<br>
<sub><!--[if gte vml 1]><v:shape id=3D"_x0000_i1054" type=3D"#_x0000_t75" s=
tyle=3D'width:132pt;
height:48pt' o:ole=3D"">
<v:imagedata src=3D"matist_sorular_dosyalar/image067.wmz" o:title=3D""/>
</v:shape><![endif]--><![if !vml]><img width=3D176 height=3D64
src=3D"matist_sorular_dosyalar/image068.gif" v:shapes=3D"_x0000_i1054"><![e=
ndif]></sub><!--[if gte mso 9]><xml>
<o:OLEObject Type=3D"Embed" ProgID=3D"Equation.3" ShapeID=3D"_x0000_i1054"
DrawAspect=3D"Content" ObjectID=3D"_1198912122">
</o:OLEObject>
</xml><![endif]--></p>
<p class=3DMsoNormal style=3D'margin-left:25.5pt;text-indent:-25.5pt;line-h=
eight:
150%;mso-list:l13 level1 lfo1;tab-stops:list 25.5pt'><![if !supportLists]><=
span
style=3D'mso-list:Ignore'>81.<span style=3D'font:7.0pt "Times New Roman"'>&=
nbsp;
</span></span><![endif]>Bir X tesadüfi değişkeninin
olasılık yoğunluk fonksiyonu verildiğinde Y=3D3-X
tesadüfi değişkeninin dağılım fonksiyonunu
değişken değiştirme tekniği ile bulunuz. <sub><!--=
[if gte vml 1]><v:shape
id=3D"_x0000_i1065" type=3D"#_x0000_t75" style=3D'width:111pt;height:41.25=
pt' o:ole=3D"">
<v:imagedata src=3D"matist_sorular_dosyalar/image069.wmz" o:title=3D""/>
</v:shape><![endif]--><![if !vml]><img width=3D148 height=3D55
src=3D"matist_sorular_dosyalar/image070.gif" v:shapes=3D"_x0000_i1065"><![e=
ndif]></sub><!--[if gte mso 9]><xml>
<o:OLEObject Type=3D"Embed" ProgID=3D"Equation.2" ShapeID=3D"_x0000_i1065"
DrawAspect=3D"Content" ObjectID=3D"_1198912123">
</o:OLEObject>
</xml><![endif]--></p>
<p class=3DMsoNormal style=3D'margin-left:25.5pt;text-indent:-25.5pt;line-h=
eight:
150%;mso-list:l13 level1 lfo1;tab-stops:list 25.5pt'><![if !supportLists]><=
span
style=3D'mso-list:Ignore'>82.<span style=3D'font:7.0pt "Times New Roman"'>&=
nbsp;
</span></span><![endif]>X tesadüfi değişkeninin
olasılık yoğunluk fonksiyonu aşağıda
verilmiştir. Y=3D<span class=3DSpellE>e<sup>X</sup></span> fonksiyonun=
un
olasılık yoğunluk fonksiyonunu değişken
değiştirme tekniği ile bulunuz. <sub><!--[if gte vml 1]><v:s=
hape
id=3D"_x0000_i1055" type=3D"#_x0000_t75" style=3D'width:111pt;height:45.75=
pt' o:ole=3D"">
<v:imagedata src=3D"matist_sorular_dosyalar/image071.wmz" o:title=3D""/>
</v:shape><![endif]--><![if !vml]><img width=3D148 height=3D61
src=3D"matist_sorular_dosyalar/image072.gif" v:shapes=3D"_x0000_i1055"><![e=
ndif]></sub><!--[if gte mso 9]><xml>
<o:OLEObject Type=3D"Embed" ProgID=3D"Equation.3" ShapeID=3D"_x0000_i1055"
DrawAspect=3D"Content" ObjectID=3D"_1198912124">
</o:OLEObject>
</xml><![endif]--></p>
<p class=3DMsoNormal style=3D'margin-left:25.5pt;text-indent:-25.5pt;line-h=
eight:
150%;mso-list:l13 level1 lfo1;tab-stops:list 25.5pt'><![if !supportLists]><=
span
style=3D'mso-list:Ignore'>83.<span style=3D'font:7.0pt "Times New Roman"'>&=
nbsp;
</span></span><![endif]><sub><!--[if gte vml 1]><v:shape id=3D"_x0000_i1056"
type=3D"#_x0000_t75" style=3D'width:231pt;height:30.75pt' o:ole=3D"">
<v:imagedata src=3D"matist_sorular_dosyalar/image025.wmz" o:title=3D""/>
</v:shape><![endif]--><![if !vml]><img width=3D308 height=3D41
src=3D"matist_sorular_dosyalar/image026.gif" v:shapes=3D"_x0000_i1056"><![e=
ndif]></sub><!--[if gte mso 9]><xml>
<o:OLEObject Type=3D"Embed" ProgID=3D"Equation.3" ShapeID=3D"_x0000_i1056"
DrawAspect=3D"Content" ObjectID=3D"_1198912125">
</o:OLEObject>
</xml><![endif]--><span style=3D'mso-spacerun:yes'> </span>göre Z=
=3DX/Y
tesadüfi değişkeninin dağılım fonksiyonunu
belirleyiniz.</p>
<p class=3DMsoNormal style=3D'margin-left:25.5pt;text-indent:-25.5pt;line-h=
eight:
150%;mso-list:l13 level1 lfo1;tab-stops:list 25.5pt'><![if !supportLists]><=
span
style=3D'mso-list:Ignore'>84.<span style=3D'font:7.0pt "Times New Roman"'>&=
nbsp;
</span></span><![endif]>X ve Y istatistiksel olarak bağımsız
tesadüfi değişkenler olup olasılık yoğunluk
fonksiyonları aşağıda verilmiştir. Z=3DX<sup>2</su=
p>Y
tesadüfi değişkeninin olasılık yoğunluk
fonksiyonunu yazınız. <a name=3D"OLE_LINK1"><sub><!--[if gte vml =
1]><v:shape
id=3D"_x0000_i1057" type=3D"#_x0000_t75" style=3D'width:228.75pt;height:54=
pt' o:ole=3D"">
<v:imagedata src=3D"matist_sorular_dosyalar/image073.wmz" o:title=3D""/>
</v:shape><![endif]--><![if !vml]><img width=3D305 height=3D72
src=3D"matist_sorular_dosyalar/image074.gif" v:shapes=3D"_x0000_i1057"><![e=
ndif]></sub><!--[if gte mso 9]><xml>
<o:OLEObject Type=3D"Embed" ProgID=3D"Equation.3" ShapeID=3D"_x0000_i1057"
DrawAspect=3D"Content" ObjectID=3D"_1198912126">
</o:OLEObject>
</xml><![endif]--></a></p>
<p class=3DMsoNormal style=3D'margin-left:25.5pt;text-indent:-25.5pt;line-h=
eight:
150%;mso-list:l13 level1 lfo1;tab-stops:list 25.5pt'><![if !supportLists]><=
span
style=3D'mso-list:Ignore'>85.<span style=3D'font:7.0pt "Times New Roman"'>&=
nbsp;
</span></span><![endif]>X ve Y istatistiksel olarak bağımsız
tesadüfi değişkenler olup olasılık yoğunluk
fonksiyonları aşağıda verilmiştir. Z=3DX<sup>2</su=
p>/Y
tesadüfi değişkeninin olasılık yoğunluk
fonksiyonunu yazınız. <sub><!--[if gte vml 1]><v:shape id=3D"_x00=
00_i1058"
type=3D"#_x0000_t75" style=3D'width:228.75pt;height:54pt' o:ole=3D"">
<v:imagedata src=3D"matist_sorular_dosyalar/image073.wmz" o:title=3D""/>
</v:shape><![endif]--><![if !vml]><img width=3D305 height=3D72
src=3D"matist_sorular_dosyalar/image074.gif" v:shapes=3D"_x0000_i1058"><![e=
ndif]></sub><!--[if gte mso 9]><xml>
<o:OLEObject Type=3D"Embed" ProgID=3D"Equation.3" ShapeID=3D"_x0000_i1058"
DrawAspect=3D"Content" ObjectID=3D"_1198912127">
</o:OLEObject>
</xml><![endif]--></p>
<p class=3DMsoNormal style=3D'margin-left:25.5pt;text-indent:-25.5pt;line-h=
eight:
150%;mso-list:l13 level1 lfo1;tab-stops:list 25.5pt'><![if !supportLists]><=
span
style=3D'mso-list:Ignore'>86.<span style=3D'font:7.0pt "Times New Roman"'>&=
nbsp;
</span></span><![endif]>X ve Y bağımsız tesadüfi
değişkenlerinin olasılık yoğunluk fonksiyonlar =
5;
h(x)=3D2x, 0<span style=3D'font-family:Symbol;mso-ascii-font-family:"Times =
New Roman";
mso-hansi-font-family:"Times New Roman";mso-char-type:symbol;mso-symbol-fon=
t-family:
Symbol'><span style=3D'mso-char-type:symbol;mso-symbol-font-family:Symbol'>=
£</span></span>x<span
style=3D'font-family:Symbol;mso-ascii-font-family:"Times New Roman";mso-han=
si-font-family:
"Times New Roman";mso-char-type:symbol;mso-symbol-font-family:Symbol'><span
style=3D'mso-char-type:symbol;mso-symbol-font-family:Symbol'>£</span>=
</span>1
ve g(y)=3Dy/8, 0<span style=3D'font-family:Symbol;mso-ascii-font-family:"Ti=
mes New Roman";
mso-hansi-font-family:"Times New Roman";mso-char-type:symbol;mso-symbol-fon=
t-family:
Symbol'><span style=3D'mso-char-type:symbol;mso-symbol-font-family:Symbol'>=
£</span></span>y<span
style=3D'font-family:Symbol;mso-ascii-font-family:"Times New Roman";mso-han=
si-font-family:
"Times New Roman";mso-char-type:symbol;mso-symbol-font-family:Symbol'><span
style=3D'mso-char-type:symbol;mso-symbol-font-family:Symbol'>£</span>=
</span>4
olduğuna göre Z=3DX<span style=3D'font-family:Symbol;mso-ascii-fo=
nt-family:
"Times New Roman";mso-hansi-font-family:"Times New Roman";mso-char-type:sym=
bol;
mso-symbol-font-family:Symbol'><span style=3D'mso-char-type:symbol;mso-symb=
ol-font-family:
Symbol'>×</span></span>Y tesadüfi değişkeninin
olasılık yoğunluk fonksiyonunu bulunuz.</p>
<p class=3DMsoNormal style=3D'margin-left:25.5pt;text-indent:-25.5pt;line-h=
eight:
150%;mso-list:l13 level1 lfo1;tab-stops:list 25.5pt'><![if !supportLists]><=
span
style=3D'mso-list:Ignore'>87.<span style=3D'font:7.0pt "Times New Roman"'>&=
nbsp;
</span></span><![endif]>Aşağıda verilen ortak olasıl=
05;k
yoğunluk fonksiyonu için (Z, W) iki boyutlu sürekli
tesadüfi değişkeninin ortak olasılık yoğunluk
fonksiyonunu çıkarınız. <span class=3DSpellE>Z’=
nin</span>
marjinal olasılık yoğunluk fonksiyonu hangi teorik
dağılımdır?
.................................................. Z=3DX<sup>2</sup>, W=3DX=
Y <sub><!--[if gte vml 1]><v:shape
id=3D"_x0000_i1059" type=3D"#_x0000_t75" style=3D'width:147pt;height:30.75=
pt' o:ole=3D"">
<v:imagedata src=3D"matist_sorular_dosyalar/image075.wmz" o:title=3D""/>
</v:shape><![endif]--><![if !vml]><img width=3D196 height=3D41
src=3D"matist_sorular_dosyalar/image076.gif" v:shapes=3D"_x0000_i1059"><![e=
ndif]></sub><!--[if gte mso 9]><xml>
<o:OLEObject Type=3D"Embed" ProgID=3D"Equation.3" ShapeID=3D"_x0000_i1059"
DrawAspect=3D"Content" ObjectID=3D"_1198912128">
</o:OLEObject>
</xml><![endif]--></p>
<p class=3DMsoNormal style=3D'margin-left:25.5pt;text-indent:-25.5pt;line-h=
eight:
150%;mso-list:l13 level1 lfo1;tab-stops:list 25.5pt'><![if !supportLists]><=
span
style=3D'mso-list:Ignore'>88.<span style=3D'font:7.0pt "Times New Roman"'>&=
nbsp;
</span></span><![endif]>Aşağıda verilen ortak olasıl=
05;k
yoğunluk fonksiyonu için (Z,W) iki boyutlu sürekli
tesadüfi değişkeninin ortak olasılık yoğunluk
fonksiyonunu çıkarınız. Z=3DY-X, W=3DX <sub><!--[if g=
te vml 1]><v:shape
id=3D"_x0000_i1060" type=3D"#_x0000_t75" style=3D'width:192pt;height:30.75=
pt' o:ole=3D"">
<v:imagedata src=3D"matist_sorular_dosyalar/image077.wmz" o:title=3D""/>
</v:shape><![endif]--><![if !vml]><img width=3D256 height=3D41
src=3D"matist_sorular_dosyalar/image078.gif" v:shapes=3D"_x0000_i1060"><![e=
ndif]></sub><!--[if gte mso 9]><xml>
<o:OLEObject Type=3D"Embed" ProgID=3D"Equation.3" ShapeID=3D"_x0000_i1060"
DrawAspect=3D"Content" ObjectID=3D"_1198912129">
</o:OLEObject>
</xml><![endif]--></p>
<p class=3DMsoNormal style=3D'margin-left:25.5pt;text-indent:-25.5pt;line-h=
eight:
150%;mso-list:l13 level1 lfo1;tab-stops:list 25.5pt'><![if !supportLists]><=
span
style=3D'mso-list:Ignore'>89.<span style=3D'font:7.0pt "Times New Roman"'>&=
nbsp;
</span></span><![endif]>f(x<sub>1</sub>,x<sub>2</sub>)=3De<sup>-(X</sup><su=
b>1</sub><sup>+X</sup><sub>2</sub><sup>)</sup><span
style=3D'mso-spacerun:yes'> </span>;x<sub>1</sub>&g=
t;0, x<sub>2</sub>>0
ise Y=3DX<sub>1</sub>/(X<sub>1</sub>+X<sub>2</sub>) tesadüfi
değişkeninin olasılık yoğunluk fonksiyonunu
değişken değiştirme tekniğini kullanarak belirleyi=
niz.
(tanım aralıklarını yazmayı unutmayınız)=
</p>
</div>
</body>
</html>
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