Expectation of Erlang Distribution

Theorem

Let $k$ be a strictly positive integer.

Let $\lambda$ be a strictly positive real number.

Let $X$ be a continuous random variable with an Erlang distribution with parameters $k$ and $\lambda$.

Then the expectation of $X$ is given by:

$\expect X = \dfrac k \lambda$

Proof

\(\ds \expect X\)

\(=\)

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

Raw Moment of Erlang Distribution

\(\ds \)

\(=\)

\(\ds \frac k \lambda\)

$\blacksquare$