Central Moment of Exponential Distribution
Jump to navigation
Jump to search
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:
- $\ds \mu_n = n! \beta^n \sum_{k \mathop = 0}^n \frac {\paren {-1}^k} {k!}$
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:
\(\ds \mu_2\) | \(=\) | \(\ds \expect {\sum_{k \mathop = 0}^n \binom n k X^{n - k} \paren {-\beta}^k}\) | Binomial Theorem | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{k \mathop = 0}^n \binom n k \paren {-\beta}^k \expect {X^{n - k} }\) | Expectation is Linear | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{k \mathop = 0}^n \beta^k \binom n k \paren {-1}^k \paren {\paren {n - k}! \beta^{n - k} }\) | Raw Moment of Exponential Distribution | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{k \mathop = 0}^n \beta^n \paren {\frac {n!} {k! \paren {n - k}!} } \paren {-1}^k \paren {n - k}!\) | Definition of Binomial Coefficient | |||||||||||
\(\ds \) | \(=\) | \(\ds n! \beta^n \sum_{k \mathop = 0}^n \frac {\paren {-1}^k} {k!}\) |
$\blacksquare$