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 $S$?
What is the distribution of the sum of n binary random variables with different probabilities and payoffs each?
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probability
statistics
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0Yes, of course that is what I meant. – 2012-08-02
1 Answers
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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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0By 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. – 2012-08-02