Moment in terms of Moment Generating Function

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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
\(\ds \) \(=\) \(\ds \frac {\d^n} {\d t^n} \sum_{m \mathop = 0}^\infty \expect {\frac {t^m X^m} {m!} }\) Expectation is Linear
\(\ds \) \(=\) \(\ds \sum_{m \mathop = 0}^\infty \frac {\d^n} {\d t^n} \paren {\frac {t^m} {m!} } \expect {X^m}\) Expectation is Linear, 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.

$\blacksquare$


Sources

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