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Chapter 6: Q. 6.42 (page 274)

The joint density of X and Y is

f(x,y)=cx2-y2e-x0鈮�x<鈭�,-x鈮�y鈮�x

Find the conditional distribution of Y, given X = x.

Short Answer

Expert verified

For t鈮�x:

P(Y鈮�t鈭�X=x)=32x3x2(t+x)-t33+x33

For t<-x:

P(Y鈮�t鈭�X=x)=0

For t>x:

P(Y鈮�t鈭�X=x)=1

Step by step solution

01

Given information

Joint density of X and Y

f(x,y)=cx2-y2e-x0鈮�x<鈭�,-x鈮�y鈮�x

also X=x

Formula to be used :

f(y鈭�x)=f(x,y)f(x)

02

Explanation

鈭�f(x,y)dy=鈭�-xxcx2-y2e-xdx=2c3x3e-x,0<x<鈭�Thus,f(y鈭�x)=32x3x2-y2,-x鈮�y鈮�x

By integration,

For t鈮�x,

localid="1647240753725" role="math" P(Y鈮�t鈭�X=x)=鈭�-xtf(x,y)dy=32x3x2(t+x)-t33+x33

Then for t < -x,

P(Y鈮�t鈭�X=x)=0

and for t > x,

P(Y鈮�t鈭�X=x)=1

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