Expectation of function in a lognormal distribution

Question

I have the function Y = e^X, which is a log-normal distribution. I am supposed to find its expectation in terms of mu and sigma. Would definitely appreciate being helped through this as I'm quite stumped!

Answer 1

I presume you mean that $X$ is normal with mean $\mu$ and standard deviation $\sigma$, and you want $\mathbb E[Y]$ where $Y = e^X$. We can write $X = \mu + \sigma Z$ where $Z$ has the standard normal distribution (mean $0$ and standard deviation $1$). Thus $$ \mathbb E[Y] = \mathbb E[e^{\mu + \sigma Z}] = e^\mu \mathbb E[e^{\sigma Z}]$$ Now either look up the Moment generating function of the standard normal distribution, or do the integration $$ \frac{e^\mu}{\sqrt{2\pi}} \int_{-\infty}^\infty e^{\sigma z} e^{-z^2/2}\; dz$$ by completing the square in the exponential.