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

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 $S$?

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Are the random variables independent? – Dilip Sarwate Aug 2 '12 at 18:28
Yes, you can assume independence – George Pastrana Aug 2 '12 at 18:34
The variable $S$ is a discrete random variable, based on your description. Hence, the concept of probability density is not applicable. Do you mean point mass function, maybe? – Sasha Aug 2 '12 at 18:46
Yes, of course that is what I meant. – George Pastrana Aug 2 '12 at 19:41

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.

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By the way, I am familiar with several papers that treat the case of Poisson Binomial distribution (as in Wikepedia article and the references contained therin). This is a somewhat more general problem where I am considering different payoffs as well as different probabilities. – George Pastrana Aug 2 '12 at 19:46