Re: random numbers from a lognormal distribution

From: Dave (dave_w_at_yahoo.com)
Date: 02/02/05 

Date: Wed, 2 Feb 2005 14:06:04 +0000 (UTC)

On Tue, 01 Feb 2005 20:33:15 -0600, Ray Koopman wrote:
>Bob Wheeler wrote:
>> [...]
>> I think he is asking how to do it himself. If X is a standard
normal variate, then U=(log(X)-m)/s is lognormal with mean
exp(m)sqrt(w), where w=exp(s^2), and variance exp(2m)w(w-1).
>
>Shouldn't that be "If X is a standard normal variate then
U=exp(X*s+m) is lognormal ..."?

This confusion reminds me of a trick I figured out for working with
the lognormal distribution...

I always had trouble working with the formulas for converting the mean
and standard deviation from the normal distribution to the lognormal
distribution, and vice versa, E(x)-->E(lnx), V(x)-->V(lnx) and so
forth. One day it occurred to me that both distributions are
completely characterized by 2 pieces of information, and that the mean
and standard deviation aren't the only 2 pieces of information I could
use. They're the traditional pieces of information, but they're not
the only ones.

I discovered that the median and interquartile range are much easier
to use, so that's what I use now.

The transformation functions exp(x) and ln(x) are non-decreasing
functions, and for any non-decreasing function g(x), the percentiles
of x and g(x) are easy to derive from each other. If x_p is the p-th
percentile of the random variable x, then g(x_p) is the p-th
percentile of the random variable g(x).

In particular, if m is the median of a normal distribution, then ln(m)
is the median of the corresponding lognormal distribution. And if m
is the median of a lognormal distribution, then exp(m) is the median
of the corresponding normal distribution. The same goes for the 25-th
and 75-th percentiles, and that allows me to calculate the
interquartile ranges of both distributions fairly easily.

So now whenever I work with the lognormal distribution, I use the
median and interquartile range as my 2 pieces of information instead
of the mean and standard deviation. Try it. I like it much better.

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