Raw Moment of Erlang Distribution
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Theorem
Let $n, k$ be strictly positive integers.
Let $\lambda$ be a strictly positive real number.
Let $X$ have a continuous random variable with an Erlang distribution with parameters $k$ and $\lambda$.
Then the $n$th raw moment of $X$ is given by:
- $\ds \expect {X^n} = \frac 1 {\lambda^n} \prod_{m \mathop = 0}^{n - 1} \paren {k + m}$
Proof
From the definition of the Erlang distribution, $X$ has probability density function:
- $\map {f_X} x = \dfrac {\lambda^k x^{k - 1} e^{- \lambda x} } {\map \Gamma k}$
From the definition of the expected value of a continuous random variable:
- $\ds \expect {X^n} = \int_0^\infty x^n \map {f_X} x \rd x$
So:
\(\ds \expect {X^n}\) | \(=\) | \(\ds \frac {\lambda^k} {\map \Gamma k} \int_0^\infty x^{n + k - 1} e^{- \lambda x} \rd x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\lambda^k} {\lambda \map \Gamma k} \int_0^\infty \paren {\frac u \lambda}^{n + k - 1} e^{- u} \rd u\) | substituting $u = \lambda x$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\lambda^k} {\lambda^{n + k} \map \Gamma k} \int_0^\infty u^{n + k - 1} e^{- u} \rd u\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {\lambda^n \map \Gamma k} \map \Gamma {n + k}\) | Definition of Gamma Function | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {\lambda^n} \frac {\map \Gamma k} {\map \Gamma k} \prod_{m \mathop = 0}^{n - 1} \paren {k + m}\) | Gamma Difference Equation | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {\lambda^n} \prod_{m \mathop = 0}^{n - 1} \paren {k + m}\) |
$\blacksquare$