I have the mean (u) and the standard deviation (sd) of a continuous distribution (X). How do I solve for the mean (u_log) and standard deviation (sd_log) of the log of that continuous distribution (log(X))?
I am looking to draw random numbers from a log-normal distribution. The problem is, most statistical packages have a log-normal-random-number-generator function that takes the mean of log(X) and sd of log(X) as inputs.
I only have the mean of X and std of X.
How to I solve for mean(log(X)) and sd(log(X)) given those inputs.
Best Answer
Let m and s be the mean and sd of $X$ on the original scale. The appropriate mean and sd on the log scale can be found after a little algebra to be
$E(\log(X)) = \log(m) - \frac{1}{2} \log [ (s/m)^2 +1]$
$sd(\log(X)) = \sqrt{\log [(s/m)^2 +1]}$