Raw Moment of Erlang Distribution

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:


 * $\displaystyle \expect {X^n} = \frac 1 {\lambda^n} \prod_{m \mathop = 0}^{n - 1} \paren {k + m}$

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:


 * $\displaystyle \expect {X^n} = \int_0^\infty x^n \map {f_X} x \rd x$

So: