# Math Help - Proving expected value of lognormal distribution

1. ## Proving expected value of lognormal distribution

How would you prove the expected value of a lognormal $(\mu, \sigma)$ distribution:

$E[X]=e^{\mu+\frac{1}{2}\sigma^2}$

I was thinking integrating $\displaystyle\int_{-\infty}^{\infty}xf(x) \ dx$ but it seems too long.

2. Originally Posted by acevipa
How would you prove the expected value of a lognormal $(\mu, \sigma)$ distribution:

$E[X]=e^{\mu+\frac{1}{2}\sigma^2}$

I was thinking integrating $\displaystyle\int_{-\infty}^{\infty}xf(x) \ dx$ but it seems too long.
It's not as long as you think, so try it. You will not infact need to eveluate an integral as after a bit of manipulation you will have something times the integral of the Gaussian between $\pm \infty$

Also the lower limit of integration is $0$ not $-\infty$

CB