Question

Checking my limits of integration for conditional expectation

Answer 1

Let (X,Y) have joint pdf \[f(x,y)=e^{-y}\]for 0<x<y, and 0 otherwise.
a) Compute marginal density of Y (simple integral, comes out to y*e^(-y) when 0<x<y)
b) Show that \[f_{X|Y}(x,y)=1/y\]again, just plain division, so that was simple.
c) Find: \[E(X|Y=y)=\int_{-\infty}^{\infty}x*f_{X|Y}(x|y)dx=\int_0^yx*\frac{1}{y}dx\]
because x is defined between 0 and y for non zero pdf.

Answer 2

Are my integration limits correct for part c? Just posted parts a and b for background.

Answer 3

@ganeshie8

Answer 4

@mathmale @agent0smith