What is the distribution of the sum of n binary random variables with different probabilities and payoffs each?

Question

Specifically, you can assume we have n random variables $X_i$ ($i \in \{1,2,3,\ldots,n\}$). Each $X_i$ has a probability $P_i$ to payoff $\mathrm{UP}_i$ and probability $Q_i=1-P_i$ to payoff $\mathrm{DOWN}_i$. $S= \sum_i X_i$. What is the probability density and cumulative distribution of $S$?

Answer 1

Denote up and down states $\{u_i\},\{d_i\}$ for convenience.

In the most generic case, where all $u_i$ and $d_i$ are different, you will have $2^n$ states $X$ of the form

$d_1 + d_2 + ... + d_n,\\ d_1 + d_2 + ... + u_n,\\ d_1 + d_2 + ... + d_{n-2} + u_{n-1} + d_n,\\ d_1 + d_2 + ... + d_{n-2} + u_{n-1} + u_n,\\ \ldots$

with the respected probabilities of

$q_1 * q_2 * ... * q_n,\\ q_1 * q_2 * ... * q_{n-1} * p_n,\\ q_1 * q_2 * ... * p_{n-1} * q_n,\\ q_1 * q_2 * ... * q_{n-1} * p_{n-1} * p_n,\\ \ldots$

and the $X$ would be the support for the pdf. Then you can get the cdf by taking cumulative partial sums of the probabilities, after sorting $X$ in increasing order.