Moment Generating Function of Binomial Distribution
Jump to navigation
Jump to search
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 Binomial 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:
- $\ds \map {M_X} t = \expect {e^{t X} } = \sum_{k \mathop = 0}^n \map \Pr {X = k} e^{t k}$
So:
\(\ds \map {M_X} t\) | \(=\) | \(\ds \sum_{k \mathop = 0}^n \binom n k p^k \paren {1 - p}^{n - k} e^{t k}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{k \mathop = 0}^n \binom n k \paren {p e^t}^k \paren {1 - p}^{n - k}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {1 - p + p e^t}^n\) | Binomial Theorem |
$\blacksquare$
Also presented as
The moment generating function $M_X$ of a binomial distribution $X$ can also be presented as:
- $\map {M_X} t = \paren {1 + p \paren {e^t - 1} }^n$
Sources
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Appendix $13$: Probability distributions
- 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): Appendix $15$: Probability distributions