Given $Y$ is a poisson random variable with mean $\theta$: $P(Y = y) = \frac{\theta^y}{y!}e^{-\theta} , y \in [0, 1, 2, 3...]$.
Moreover, $Var(Y) = \theta$ and $X = Y + Z$ where $Z$ is Bernoulli random variable with expectation $\frac{1}{2}$. $P(Z = 1) = \frac{1}{2}$ and $P(Z = 0) = \frac{1}{2}$ and $Z$ is independent of $Y$. I need to find $E[X^2\mid Y]$.
I tried to solve this problem using convolution of p.m.f since I known $Y$ and $Z$ are independent and have their corresponding p.m.f. But I can only find p.m.f of $X$ and this does not seem to help me find the conditional expectation. Any ideas?
You have $$E\,[X^2\mid Y=y]=E\,[(Y+Z)^2\mid Y=y]=g(y)\quad,(\text{say})$$
By linearity of expectation,
$$g(y)=E\,[Y^2\mid Y=y]+E\,[2YZ\mid Y=y]+E\,[Z^2\mid Y=y]$$
Knowing that $Z$ (and any function of $Z$) is independent of $Y$, the above can be simplified.
Your final answer is $g(Y)$.