# 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 $\ds$ $=$ $\ds \frac k \lambda$

$\blacksquare$