Variance 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 variance of $X$ is given by:


 * $\var X = \dfrac k {\lambda^2}$

Proof
By Variance as Expectation of Square minus Square of Expectation, we have:


 * $\var X = \expect {X^2} - \paren {\expect X}^2$

By Expectation of Erlang Distribution, we have:


 * $\expect X = \dfrac k \lambda$

We also have:

So: