# Moment in terms of Moment Generating Function

Jump to navigation
Jump to search

## Theorem

Let $X$ be a random variable.

Let $M_X$ be the moment generating function of $X$.

Then:

- $\expect {X^n} = \map { {M_X}^{\paren n} } 0$

where:

- $n$ is a non-negative integer
- ${M_X}^{\paren n}$ denotes the $n$th derivative of $M_X$
- $\expect {X^n}$ denotes the expectation of $X^n$.

## Proof

\(\ds \map { {M_X}^{\paren n} } t\) | \(=\) | \(\ds \frac {\d^n} {\d t^n} \expect {e^{t X} }\) | Definition of Moment Generating Function | |||||||||||

\(\ds \) | \(=\) | \(\ds \frac {\d^n} {\d t^n} \expect {\sum_{m \mathop = 0}^\infty \frac {t^m X^m} {m!} }\) | Power Series Expansion for Exponential Function | |||||||||||

\(\ds \) | \(=\) | \(\ds \frac {\d^n} {\d t^n} \sum_{m \mathop = 0}^\infty \expect {\frac {t^m X^m} {m!} }\) | Linearity of Expectation Function | |||||||||||

\(\ds \) | \(=\) | \(\ds \sum_{m \mathop = 0}^\infty \frac {\d^n} {\d t^n} \paren {\frac {t^m} {m!} } \expect {X^m}\) | Linearity of Expectation Function, Power Series is Termwise Differentiable within Radius of Convergence | |||||||||||

\(\ds \) | \(=\) | \(\ds \sum_{m \mathop = n}^\infty \frac {m^{\underline n} t^{m - n} } {m!} \expect {X^m}\) | Nth Derivative of Mth Power | |||||||||||

\(\ds \) | \(=\) | \(\ds \sum_{m \mathop = n}^\infty \frac { m! t^{m - n} } {m! \paren {m - n}!} \expect {X^m}\) | Falling Factorial as Quotient of Factorials | |||||||||||

\(\ds \) | \(=\) | \(\ds \frac {t^{n - n} } {\paren {n - n}!} \expect {X^n} + \sum_{m \mathop = n + 1}^\infty \frac {t^{m - n} } {\paren {m - n}!} \expect {X^m}\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds \expect {X^n} + \sum_{m \mathop = n + 1}^\infty \frac {t^{m - n} } {\paren {m - n}!} \expect {X^m}\) |

Setting $t = 0$ yields the result.

$\blacksquare$