Expected value of lognormal distribution.

Question

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.

score 4 · Accepted Answer · answered Dec 13 '14 at 06:34

4

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 $X$.

answered Dec 13 '14 at 06:34

heropup

135,869

Thanks so I get $e^{k\mu}e^{\frac 12 \sigma^2 k^2}$ for the expected value. – user24013423 Dec 13 '14 at 06:48
2

@user24013423 I think you want to set $k=1$ there – JessicaK Dec 13 '14 at 06:53

Expected value of lognormal distribution.

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