TITLE: How do I calculate the conditional expectation?
QUESTION [1 upvotes]: How do I find a conditional expectation, $E(Y|X)$ for:
$f_{XY}(x,y)=2e^{-(x+y)}$ for $0<x<y$.
I researched that the formula for a conditional expectation is:
$E(Y|X) = \Sigma yf_{Y|X}(y|x)$
And I have already calculated the conditional density as:
$f_{Y|X}(y|x) = e^{x-y}$ for $0<x<y$
Thus,
$E(Y|X) = \Sigma ye^{x-y}$ for $0<x<y$

REPLY [1 votes]: The conditional density you have found is for $0<x<y$. $E(Y|X)=\int_x^{\infty} ye^{x-y} dy$. Integrate by parts to find the exact value.

REPLY [0 votes]: First your density function $f_{Y\mid X=x}$ is wrong. Anyway, suppose it's true : what you computed (if you replace of cours $\sum_{y}$ by $\int_{\mathbb R}$) is $$\mathbb E[Y\mid X=x].$$
But $$\mathbb E[Y\mid X]=\int_{\mathbb R}ye^{X-y}\,\mathrm d y=e^X\int_{\mathbb R}ye^{-y}\,\mathrm d y.$$
{"set_name": "stack_exchange", "score": 1, "question_id": 3236899}