I am preparing for my test which is on Friday. There is this problem which is killing me and I don't have any idea how to solve it! It is in conditional probabilities chapter. The question is
X ~ Poi (a)
Y ~ Poi (b)
Z = X^Y
What is the Expected value of Z.
I will be so thankful if anyone could help me.
Is itOriginally Posted by Olivia787878
?
You know that, this is the moment generating function. Or maybe you know
, which is the same with
. Anyway, you can compute this, even if you don't know it already.
On the other hand, since (you should have mentioned it) X and Y are independent, the conditional distribution of Y given X=x is the same as the distribution of Y. Therefore, for all,
. Then
, and the last expectation can be computed using the moment generating function again.
The last inequalities can also be written![E[X^Y]=\sum_{x=0}^\infty E[X^Y|X=x] P(X=x) = \sum_{x=0}^\infty e^{b(x-1)} P(X=x)](http://latex.codecogs.com/png.latex?E[X^Y]=\sum_{x=0}^\infty E[X^Y|X=x] P(X=x) = \sum_{x=0}^\infty e^{b(x-1)} P(X=x))
.
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