# Conditional Expectation involving Poisson Distribution

• January 20th 2010, 07:59 AM
statmajor
Conditional Expectation involving Poisson Distribution
Let X1,...,Xn be a random sample from a Poisson distribution with mean theta. Find $E(Y_1 = X_1 + 2X_2 + 3X_3 | Y_2 = \Sigma X_i)$

I know that $\Sigma X_i = Pois(\theta n)$

I need to find $P(Y_1|Y_2) = \frac{P(Y_1,Y_2)}{P(Y_2)}$

How would I find the joint PMF of Y1 and Y2?
• January 20th 2010, 08:51 AM
Moo
Hello,

I think the formula you use is only available with probability distributions with densities.

Anyway, look here : http://www.mathhelpforum.com/math-he...757-post2.html

which proves that $E[X_i|Y_2]=\frac{Y_2}{3}~,~\forall i \in\{1,2,3\}$

And then just write $E[Y_1|Y_2]=E[X_1|Y_2]+2E[X_2|Y_2]+3E[X_3|Y_2]$ (linearity of the expectation)

And you're done.