Variance of Log Normal Distribution

Theorem
Let $X$ be a continuous random variable with the Log Normal distribution with $\mu \in \R, \sigma \in \R_{> 0}$.

Then the variance of $X$ is given by:


 * $\var X = \exp {\paren {2\mu + \sigma^2 } } \paren {\exp {\paren {\sigma^2  } } - 1}$

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 Log Normal Distribution, we have:


 * $\expect X = \exp {\paren {\mu + \dfrac {\sigma^2 } 2 } }$

From Raw Moment of Log Normal Distribution, we have:

The $n$th raw moment $\expect {X^n}$ of $X$ is given by:


 * $\expect {X^n} = \exp {\paren {n\mu + \dfrac {\sigma^2 n^2 } 2 } }$

Therefore, for $n = 2$ we have:


 * $\expect {X^2} = \exp {\paren {2\mu + 2\sigma^2 } }$

So: