# Thread: The conditional properties of 2 exponential functions

1. ## The conditional properties of 2 exponential functions

Suppose $\displaystyle f_{X|Y}(x|y)=\frac{1}{y}e^{-\frac{x}{y}}$ and $\displaystyle f_Y(y)=e^{-y}$ where $\displaystyle 0<x,y<\infty$

Why is it obvious that $\displaystyle E(X|Y)=Y$ and $\displaystyle Var(X|Y)=Y^2$ ??

2. Hello,

The distribution of X|Y is that of an exponential distribution with parameter 1/Y. So its expectation will be 1/(1/Y)=Y and the variance...