Two random variables follow the following Binomial distribution.
$N_i\sim B(S_i,\delta)$, $H_i \sim B(N_i,\alpha)$, where $\alpha$ is not a function of $\delta$.
These two random variables are related in a sense that in each period i, $N_i$ is determined first then realized $N_i$ gets into $B(N_i,\alpha)$.
I am interested in calculating the following: $$E\left[\frac{\sum_{i=1}^T H_i}{\sum_{i=1}^T N_i}\right]$$
Let's assume that $N_1=1$ so that we don't consider the case where the denominator becomes zero. Intuitively, this expected value would be alpha. However, how can I prove this mathematically?