Raw Moment of Exponential Distribution
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Theorem
Let $X$ be a continuous random variable of the exponential distribution with parameter $\beta$ for some $\beta \in \R_{>0}$.
Let $n$ be a strictly positive integer.
Then the $n$th raw moment of $X$ is given by:
- $\expect {X^n} = n! \beta^n$
Proof
From Moment Generating Function of Exponential Distribution, the moment generating function of $X$ is given by:
- $\map {M_X} t = \dfrac 1 {1 - \beta t}$
By Moment in terms of Moment Generating Function:
- $\expect {X^n} = \map {M^{\paren n}_X} 0$
We have:
\(\ds \map {M^{\paren n}_X} t\) | \(=\) | \(\ds \frac {\d^n} {\d t^n} \paren {\frac 1 {1 - \beta t} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds -\frac 1 \beta \frac {\d^n} {\d t^n} \paren {\frac 1 {t - \frac 1 \beta} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds -\frac {\paren {-1}^n n!} {\beta \paren {t - \frac 1 \beta}^{n + 1} }\) | Nth Derivative of Reciprocal of Mth Power: Corollary, Chain Rule for Derivatives | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\paren {-1}^{n + 1} n!} {\frac \beta {\beta^{n + 1} } \paren {\beta t - 1}^{n + 1} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\paren {-1}^{n + 1} n! \beta^n} {\paren {\beta t - 1}^{n + 1} }\) |
Setting $t = 0$ gives:
\(\ds \expect {X^n}\) | \(=\) | \(\ds \frac {\paren {-1}^{n + 1} n! \beta^n} {\paren {-1}^{n + 1} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds n! \beta^n\) |
$\blacksquare$