# Math Help - Conditional expectation, distribution

1. ## Conditional expectation, distribution

X and Y are independent, X~P(5), Y has Bernoulli's distribution with P(Y=1)=0.3. How to find distribution of E[XY|X]?
XY and Y are not independent so I cant write it as EXEY.. So probably I should use the fact that X~P(5) and P(Y=1)=0.3.. But dont know how to do that correctly

2. ## Re: Conditional expectation, distribution

Hello,

X is X-measurable, so you can write that E[XY|X]=XE[Y|X]. And then since Y and X are independent, E[Y|X]=E[Y].
So the distribution of E[XY|X] is that of 0.3*P(5) (same probabilities, but the values of the conditional expectation are in $0.3*\mathbb{N}=\{0;0.3;0.6;0.9;\dots\}$)