Imagine I have a process with $N$ distinct trials, $(t_1,t_2,t_3,t_4,...,t_N)$, where each trial $t_i$ has its own probability of success $p_i$ and probability of failure $q_i = (1-p_i)$.
After performing a single random instance of each trial, how do we calculate something like a probability distribution for the overall number of successes, or less ambitiously, the mean / median number of expected overall successes?
For example:
Let's say we perform ten distinct trials, $(t_1,t_2,...,t_10)$, a single time per trial, where the probabilities of success for each trial are $(0.3,0.23,0.94,0.1,0.1,0.44,0.1,0.4955,0.64,0.0987)$. What probability distribution do we have for the number of overall successes, or can we at least compute a mean / median?
The probability generating function for $Y = X_1 + \ldots + X_n$ is $$E[ z^Y] = \prod_{i=1}^n E[z^{X_i}] = \prod_{i=1}^n (q_i + p_i z) $$ Expand this and you can read off the probability mass function of $Y$.
Or you can proceed iteratively: if $Y_k = X_1 + \ldots + X_k$, then for $0 \le j \le k$, $$P(Y_k = j) = q_k P(Y_{k-1} = j) + p_k P(Y_{k-1} = j-1)$$
There is no nice expression for the median, however.