The sum of independent Bernoulli random variables with possibly unequal success probabilities has a Poisson binomial distribution. Its probability generating function is : expand the product to read off the probability mass function. Linearity of expectation and additivity of variance for independent random variables give and . Equal recover the binomial distribution.
For three trials with success probabilities , expanding gives masses at . Their expected value is and variance is . Replacing all three success probabilities by their average preserves the expected value but gives binomial distribution variance , so it does not preserve the law.
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The Poisson binomial distribution is a generalization of the binomial distribution. It is used to model the number of successes in a sequence of independent Bernoulli trials, where each trial can have a different probability of success. In contrast, the binomial distribution assumes that each trial has the same probability of success. ### Key Characteristics: 1. **Independent Trials**: The trials are independent of each other.