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Chapter 7: Q.50 (page 356)

The joint density of X and Yis given by f(x,y)=e-x/ye-yy,0<x<∞,0<y<∞, Compute EX2∣Y=y.

Short Answer

Expert verified

EX2∣Y=y=2y2

Step by step solution

01

Given information

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

f(x,y)=e-x/ye-yy,0<x<∞,0<y<∞.

02

Explanation

The joint density of Xand Yis given by f(x,y)=e-xye-yywhere 0<x<∞,0<y<∞. We have to compute EX2∣Y=y.

From the definition we have:

E[X∣Y=y]=∫-∞∞xfx∣y(x∣y)dx

Therefore it simply follows:

EX2∣Y=y=∫-∞∞x2·f(x,y)fY(y)dx

=∫0∞x2·e-xye-yy∫-∞∞f(x,y)dxdx

=∫0∞x2·e-xye-yye-ydx

=1y∫0∞x2·e-xydx

=1y·2y3=2y2

03

Final answer

EX2∣Y=y=2y2

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