Central 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 central moment $\mu_n$ of $X$ is given by:

$\displaystyle \mu_n = n! \beta^n \sum_{k \mathop = 0}^n \frac {\paren {-1}^k} {k!}$


Proof

From definition of central moment we have:

$\mu_n = \expect {\paren {x - \mu}^n}$

By Expectation of Exponential Distribution we have:

$\mu = \beta$

So:

\(\displaystyle \mu_2\) \(=\) \(\displaystyle \expect {\sum_{k \mathop = 0}^n \binom n k X^{n - k} \paren {-\beta}^k}\) Binomial Theorem
\(\displaystyle \) \(=\) \(\displaystyle \sum_{k \mathop = 0}^n \binom n k \paren {-\beta}^k \expect {X^{n - k} }\) Linearity of Expectation Function
\(\displaystyle \) \(=\) \(\displaystyle \sum_{k \mathop = 0}^n \beta^k \binom n k \paren {-1}^k \paren {\paren {n - k}! \beta^{n - k} }\) Raw Moment of Exponential Distribution
\(\displaystyle \) \(=\) \(\displaystyle \sum_{k \mathop = 0}^n \beta^n \paren {\frac {n!} {k! \paren {n - k}!} } \paren {-1}^k \paren {n - k}!\) Definition of Binomial Coefficient
\(\displaystyle \) \(=\) \(\displaystyle n! \beta^n \sum_{k \mathop = 0}^n \frac {\paren {-1}^k} {k!}\)

$\blacksquare$