# How to simulate lognormal distributions?

1. May 17, 2012

### jackaip

I am studying statistics and am interested in understanding the log normal distribution. From some discussion I gather that the log normal distributions arises from multiplicative effects while the normal distribution arises from additive effects. I generated the following matlab code to simulate the normal distribution.

clear
for j = 1:10000
S = 0;
for i = 1:1000
S = S + rand(1)-0.5;
end
T(j) = S;
end
hist(T,100)

If you add a bunch of random numbers between -0.5 and 0.5 you get more or less a gaussian
by the central limit theorem.

Now I am interested in simulating the log normal distribution, but not sure what to do. Has anyone ever tried this. I saw that if the variance grows in time in certain stock market models you can get a log normal distribution. Also the number of proteins in cells is more or less log normal distributed.

Thanks,
jackaip

2. May 17, 2012

### Stephen Tashi

If you generate $X$ as a normal random variable then $e^X$ is a lognormal random variable.

You should look at better algorithms for generating normal random variates. I don't know them by memory, but if I happen to run across one, I'll post it.

3. May 18, 2012

### chiro

Hey jackaip.

If you can simulate a normal variable (if you don't have a tool, use R: it's free and you will able to do what you need in 5 minutes), then simply simulate a normal and then calculate a new variable which is eX which basically calculates the exponential of each realization created and stores this in another vector. This vector will be log-normally distributed.

In R, this will look like:

x = rnorm(10000,0,1)
lognorm = exp(x)
hist(lognorm)

This will produce a histogram. If you want actual values just use lognorm and reference each component at location i using lognorm.

If you need to do this in MATLAB, then basically generate a simulated normal and then just apply exp(x) where x is each generated value and throw that in a vector.

4. May 18, 2012

### viraltux

Hi All!

In fact R already has the log-normal distribution implemented, so simply rlnorm will suffice to generate those samples.

And as for jackip's question, the algorithm to generate the normal distribution, in your code:

Code (Text):
for j = 1:10000
S = 0;
for i = 1:1000
S = S + rand(1)-0.5;
end
T(j) = S;
end
You are using a random walk to generate the normal distribution which is extremely inefficient as well as inaccurate, so, I guess you do this not because you don't know better but rather because this way you can see how a normal distribution is constructed and, likewise, you'd like to see again how the log normal is constructed. Am I right?

Well, if this is so you just need to change this T(j) = exp(S) in your code to converge to a log-normal distribution and, since S is a summatory, you can turn it into the following construction:

Code (Text):
for j = 1:10000
S = 1;
for i = 1:1000
S = S * exp(rand(1)-0.5);
end
T(j) = S;
end
Which is even more inefficient and inaccurate but it gives you the rough understanding on how a log normal distribution is constructed which I guess is what you're looking for. right?

Last edited: May 18, 2012