# How to find the correlation between two random numbers

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Trond Oesten on 24 Feb 2015
Commented: Greig on 25 Feb 2015
Hi,
I want to give a specific correlation between two random numbers. I am using them to find the inverse lognormal distribution, so they have to remain between 0 and 1. How can I do this?
script:
N = 10;
ro = 0.75; %correlation between random numbers
f = rand(N,2);
mu7 = 0.84208;
su7 = 0.3852;
my7 = 0.688;
sy7 = 0.09975;
j = f(:,1);
g = f(:,2);
epy = logninv(j,my7,sy7);
epu_y = logninv(g,mu7,su7);
epu = epu_y + epy;
Thanks!

Greig on 24 Feb 2015
Edited: Greig on 25 Feb 2015
If you have two uncorrelated normally distributed random numbers, given by x, you can use the following to determine Y, which will be correlated with x(:,1)....
x=randn(1e3,2);
R0 = 0.75;
corr(x(:,1), x(:,2))
Y=R0*x(:,1)+(sqrt(1-R0)*x(:,2));
corr(x(:,1), Y)
It also appears to work well if x is from a uniform distribution, but the returned results are no longer uniformly distributed.
x=rand(1e3,2);
R0 = 0.75;
corr(x(:,1), x(:,2))
Y=R0*x(:,1)+(sqrt(1-R0)*x(:,2));
corr(x(:,1), Y)
It should be noted that the real correlation between x(:,1) and Y can vary significantly from R0 if N is small, also it works less well if x(:,1) and x(:,2) are coincidently correlated.
There are ways of doing this with Cholesky and eignevector decomposition, but I can't remember them off the top of my head.
UPDATED:
As John pointed out, the about solution does not maintain the uniformity of the random numbers after adjusted for correlations. I have attached a function that I sometimes use for this purpose, which does return uniform results
This is based on an example script provided here http://comisef.wikidot.com/tutorial:correlateduniformvariates
Greig on 25 Feb 2015
The answer has been updated with a script that returns uniformly distributed results