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
\(\ds \) \(=\) \(\ds \frac 1 {\lambda^n} \prod_{m \mathop = 0}^{n - 1} \paren {k + m}\)

$\blacksquare$