Expected Value of a Lognormal Random Variable

If $\log X \sim N(0,1)$ , calculate $\mathbb{E}[X]$ .

Answer

Let $\log X = Y \sim N(0,1)$ . Then $$X = e^Y$$ , so

\mathbb{E}[X] = \mathbb{E}[e^Y] = M_Y(1),

where $$M_Y(t)$$ is the MGF of $$Y$$ .

For $Y \sim N(0,1)$ ,

M_Y(t) = \exp\!\left(\tfrac{1}{2}t^2\right).

Hence,

\mathbb{E}[X] = M_Y(1) = e^{1/2}.