How to solve expectation of continuous variables

Question

If it is assumed that $x\thicksim N(0,\sigma^2)$, then it can be shown that $E\{exp(x)\}=exp(0.5\sigma^2)$

How to show that?

Answer 1

$E\exp tX$ is called the moment-generating function or mgf of $X$. We can prove $E\exp tX=\exp \sigma^2t^2/2$ for this distribution. The expectation is this integral (note the substitution $y=x-\sigma^2 t$): $$\int_{\Bbb R}\frac{1}{\sigma\sqrt{2\pi}}\exp\bigg(tx-\frac{x^2}{2\sigma^2}\bigg)dx=\int_{\Bbb R}\frac{1}{\sigma\sqrt{2\pi}}\exp\bigg(\frac{\sigma^4t^2-y^2}{2\sigma^2}\bigg)dy=\exp \sigma^2 t^2/2.$$