Probability Generating Function of Binomial Distribution

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Theorem

Let $X$ be a discrete random variable with the binomial distribution with parameters $n$ and $p$.


Then the p.g.f. of $X$ is:

$\Pi_X \left({s}\right) = \left({q + ps}\right)^n$

where $q = 1 - p$.


Proof

From the definition of p.g.f:

$\displaystyle \Pi_X \left({s}\right) = \sum_{k \mathop \ge 0} p_X \left({k}\right) s^k$

From the definition of the binomial distribution:

$\displaystyle p_X \left({k}\right) = \binom n k p^k \left({1 - p}\right)^{n-k}$

So:

\(\displaystyle \Pi_X \left({s}\right)\) \(=\) \(\displaystyle \sum_{k \mathop = 0}^n \binom n k p^k \left({1 - p}\right)^{n - k} s^k\)
\(\displaystyle \) \(=\) \(\displaystyle \sum_{k \mathop = 0}^n \binom n k \left({p s}\right)^k \left({1 - p}\right)^{n - k}\)
\(\displaystyle \) \(=\) \(\displaystyle \left({\left({p s}\right) + \left({1 - p}\right)}\right)^n\) by the Binomial Theorem

Hence the result.

$\blacksquare$


Sources