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The Poisson Binomial Distribution is the discrete probability distribution of a sum of independent Bernoulli trials that are not necessarily identically distributed. The first four moments of this distribution are shown in the figure (source: https://en.wikipedia.org/wiki/Poisson_binomial_distribution).

Suppose we are not dealing with a series of typical Bernoulli trials. Rather, suppose we are dealing with trials, where each Bernoulli Distribution $i$ is multiplied by constant, $c_i$, such that some trials are essential weighted more than others.

What are the moments of this more general Poisson Binomial Distribution?

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A Poisson binomial distribution is a sum of $n $ independent Bernoulli distributed variables, so that by definition their moments are the sums of the moments of the $n$ Bernoulli distributions. For any value of $i $ (i.e., for any of these Bernoulli distributed variables), if $p_i $ is multiplied by a constant $c_i $, the resulting moments are simply obtained by substituting $p_i $ with $p_i \cdot c_i $. Extending this to the sums over all $n $ possible values of $i $ you can get the moments for this generalized Poisson binomial distribution.

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