Moment Generating Function of Binomial Distribution

Theorem
Let $X \sim \Binomial n p$ for some $n \in \N$ and $0 \le p \le 1$.

Then the moment generating function $M_X$ of $X$ is given by:


 * $\displaystyle \map {M_X} t = \paren {1 - p + p e^t}^n$

Proof
From the definition of the Poisson distribution, $X$ has probability mass function:


 * $\displaystyle \map \Pr {X = k} = \binom n k p^k \paren {1 - p}^{n - k}$

From the definition of a moment generating function:


 * $\displaystyle \map {M_X} t = \expect {e^{t X} } = \sum_{k \mathop = 0}^n \map \Pr {X = k} e^{t k}$

So: