What exactly is the Lognormal distribution?
Also how can I find it's distribution.
I came across the following problem in Sheldon M Ross, I am not understanding where to start. Please help
A random variable $X$ is said to have a lognormal distribution if $\log X$ is normally distributed. If $X$ is lognormal with $\mathrm{E}(\log X) = \mu$ and $\mathrm{Var}(\log X) = \sigma^2$ , determine the distribution function of $X$. That is, what is $P(X \leq x)$?
Best Answer
The RV $Y=\log(X) \sim N(\mu,\sigma^2)$.
Then for $x>0$ we have $$P(X\le x)=P(\log(X)\le\log(x))=P(Y\le \log(x))=F_{\mu,\sigma}(\log(x)),$$ where $F_{\mu,\sigma}(\cdot)$ denotes the cumulative distribution function for a Normal RV with mean $\mu$ and variance $\sigma^2$.