Let $X$ and $Y$ be two independent random variables with Poisson distribution. Also, let $F_2(y,\mu_2)$ be the Poisson CDF of $Y$ with mean $\mu_2$ and $P_1(x,\mu_1)$ be the Poisson PMF of $X$ with mean $\mu_1$. Then, what would be the distribution of $h$ which is defined as follows. Can we say $h$ has a Poisson distribution or not? If so, what's its mean?
$h=\sum_{t=\alpha-\beta+1}^{\alpha}F_2(\alpha-t,\mu_2)P_1(t,\mu_1)$