Hi I'm stuck on this question:
Recall that $X$ is said to have a lognormal distribution with parameters $\mu$ and $\sigma^2$ if $\log(X)$ is normal with mean $\mu$ and variance $\sigma^2$.
Suppose $X$ is such a lognormal random variable.
- Find $\mathrm{E}[X]$.
- Find $\mathrm{Var}(X)$.
I know that the approach is to find the moment generating function and take the first derivative for the expected value and the second derivative for the variance (I think), but I can't seem to figure it out. Thanks in advance.
No. The lognormal distribution doesn't have a moment generating function, so you can't use that approach.
Instead, suppose $Y = \log X$. Then $Y \sim \operatorname{Normal}(\mu, \sigma^2)$ by definition, and $X = e^Y$. Therefore for a positive integer $k$, $$\operatorname{E}[X^k] = \operatorname{E}[e^{kY}] = M_Y(k),$$ where $M_Y(k)$ is the moment generating function of $Y$. This hint should now make it trivial to obtain the expectation and variance of $X$.