Expectation of Log Normal Distribution

Theorem

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

The expectation of $X$ is given by:

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

Proof

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 $n = 1$ we have:

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

Hence the result.

$\blacksquare$