Q. 7.49 - A First Course in Probability

Question

The positive random variable $X$ is said to be a lognormal random variable with parameters $\mu$ and${\sigma }^2$ if $\log \left(X\right)$ is a normal random variable with mean $\mu$and variance ${\sigma }^2$. Use the normal moment generating function to find the mean and variance of a lognormal random variable 

Step-by-Step Solution

Answer

The mean and variance of a lognormal random variable value are $E\left(X\right)=e^{mu+{\sigma }^2/2}$and

Variance  $\left(X\right)=e^{2\mu +{\sigma }^2}\left(e^{{\sigma }^2}-1\right)$.

Step 1: Given Information

Use the normal moment generating function to find the mean and variance of a lognormal random variable. 

Step 2: Explanation

Define, $Y=\log X$

Since we know that$X$ is log-normal random variable with parameters $\mu$and ${\sigma }^2$, we have that $Y~N\left(\mu ,{\sigma }^2\right)$.

Observe the following equality. For $t\ge$$0$

We have that,

$E\left(X^t\right)=E\left({\left(e^Y\right)}^t\right)=E\left(e^{tY}\right)=M_Y\left(t\right)$

$=\exp \left(\mu t+\frac{{\sigma }^2{t}^2}{2}\right)$.

Step 3: Explanation

Now we have that,

$E\left(X\right)=\exp \left(\mu +\frac{{\sigma }^2}{2}\right)$and

$E\left(X^2\right)=\exp \left(2\mu +2{\sigma }^2\right)$

Which implies,  variance  $\left(X\right)=E\left(X^2\right)-E(X{)}^2$

$=\exp \left(2\mu +2{\sigma }^2\right)-\exp \left(2\mu +{\sigma }^2\right)$

$\Rightarrow Var\left(X\right)=e^{2\mu +{\sigma }^2}\left(e^{{\sigma }^2}-1\right)$.

Step 4: Final answer

The mean and variance of a lognormal random variable value are $E\left(X\right)=e^{mu+{\sigma }^2/2}$ and

variance $\left(X\right)=e^{2\mu +{\sigma }^2}\left(e^{{\sigma }^2}-1\right)$.