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 $P(X \leq x)$?

  • $\begingroup$ Do you know what it means for a random variable Y to be normally distributed? $\endgroup$ – lulu Sep 6 '15 at 9:31
  • $\begingroup$ Yes. But I am not able to find the distribution function for the lognormal distribution. $\endgroup$ – LoverOfAbstraction Sep 6 '15 at 9:53
  • $\begingroup$ But it's the same. If $Y$ is normal, with the data you provide, then the density is $$\frac {1}{\sigma \sqrt {2\pi}}e^{-\frac {(x-\mu)^2}{2\sigma ^2}}$$. Thus, in your case, the density for $LogX$ is $$\frac {1}{\sigma \sqrt {2\pi}}e^{-\frac {(Logx-\mu)^2}{2\sigma ^2}}$$ $\endgroup$ – lulu Sep 6 '15 at 10:20
  • $\begingroup$ See Wikipedia on 'log-normal distribution': The terminology and parameters can be confusing: A lognormal dist'n is a continuous probability distribution of a random variable whose logarithm is normally distributed. Thus, if the random variable $X$ is lognormally distributed, then $Y = \log(X)$ (log base $e$) has a normal distribution. Likewise, if $Y$ has a normal distribution, then $X = \exp(Y)$ has a lognormal distribution. The parameters $\mu$ and $\sigma$ are moments of the $normal$ distribution, but are often used for the lognormal distribution also. To be clear, $E(Y) = \mu$, not $E(X).$ $\endgroup$ – BruceET Sep 7 '15 at 0:19
  • $\begingroup$ @lulu, you are missing $x$ after $\sqrt{2\pi}$. $\endgroup$ – Tomasz Kania Nov 26 '17 at 15:11

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 $\sigma^2$.


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