expectation of distribution where arguments are distributions too

Question

I want to find an expectation and variance of c, where

$ c | a, b \sim Poiss(0.5 a + 0.05 b) $

$ a \sim R(15, 30) $

$ b \sim R(250, 350) $

Where R is uniform distribution, Poiss is Poisson distribution.

Should I use expectation of a and b to find expectation of c or it would be wrong?

Answer 1

Hint: You can use conditional expectations. Since $c$ is conditionally Poisson, we have $\mathbb{E}(c\mid a,b)=.5a+.05b.$ It follows that $$\mathbb{E}(c)=\mathbb{E}(\mathbb{E}(c\mid a,b))=\mathbb{E}(.5a+.05b).$$ Similarly, $$\mathbb{E}(c^2)=\mathbb{E}(\mathbb{E}(c^2\mid a,b))=\mathbb{E}\left((.5a+.05b)(.5a+.05b+1)\right).$$ We use the fact that if $X$ is Poisson($\lambda$), then $\mathbb{E}(X)=\lambda$ and $\mathbb{E}(X^2)=\lambda(\lambda+1)$.