Moment Generating Function of Binomial Distribution

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Theorem

Let $X$ be a discrete random variable with a binomial distribution with parameters $n$ and $p$ for some $n \in \N$ and $0 \le p \le 1$:

$X \sim \Binomial n p$


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

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


Proof

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

$\map \Pr {X = k} = \dbinom n k p^k \paren {1 - p}^{n - k}$

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:

\(\displaystyle \map {M_X} t\) \(=\) \(\displaystyle \sum_{k \mathop = 0}^n \binom n k p^k \paren {1 - p}^{n - k} e^{t k}\)
\(\displaystyle \) \(=\) \(\displaystyle \sum_{k \mathop = 0}^n \binom n k \paren {p e^t}^k \paren {1 - p}^{n - k}\)
\(\displaystyle \) \(=\) \(\displaystyle \paren {1 - p + p e^t}^n\) Binomial Theorem

$\blacksquare$