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Chapter 7: Q.51 (page 356)

The joint density of Xand Yis given by

f(x,y)=e-yy,0<x<y,0<y<鈭�

Compute EX3鈭�Y=y.

Short Answer

Expert verified

EX3鈭�Y=y=y34

Step by step solution

01

Given information

Given in the question that, The joint density of Xand Yis given by

f(x,y)=e-yy,0<x<y,0<y<鈭�.

02

Explanation

The joint density of Xand Yis given by f(x,y)=e-yyon the district 0<x<yand 0<y<鈭�. For simpler arrangement we'll introduce a diagram on the subsequent page. The goal is to work out the normal worth ofX3given Y=y. Initially note that:

f(x鈭�y)=f(x,y)fY(y)=e-yy路1e-y=1y

Now we simply calculate:

EX3鈭�Y=y=鈭�0yx3路1ydx=1y路x440y=y34

03

Graph

We integrate on the following region:

04

Final answer

EX3鈭�Y=y=y34

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