Prove the expected value of a normal variable

Question

I would like to ask how would I prove that if $X$ is a normal variable with mean $\mu$ and variance $\sigma^2$, then $\mathbb{E}e^{u(X-\mu)} = e^{\frac{1}{2}u^2\sigma^2}$?

Since you have not shown any effort it is hard to guess what kind of proof will be accessible to you. — Kavi Rama Murthy, Nov 12 '20 at 07:30

score 0 · Accepted Answer · answered Nov 12 '20 at 08:10

0

Since $X\sim N(\mu;\sigma^2)$ it is easy to prove that

$$Y=u(X-\mu)\sim N(0;\sigma^2 u^2)$$

Thus

$e^Y$ is lognormal distributed with mean $e^{\frac{\sigma^2 u^2}{2}}$

answered Nov 12 '20 at 08:10

tommik

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Prove the expected value of a normal variable

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