Generating a random variable that has a predefined covariance with other random variable

I wanted to generate a normal random variable which has a predefined co-variance with another random number. But I can't figure out how to proceed.Here is the specific problem,

I have e~N(0,1) and I want to generate X such that cov(x,e)=1 and E(X)=0. How can I generate this random number with sample size 500?

Thanks!

Answer 1

[Roger Stafford]

You haven't stated what you want the variance of X to be. If it is to be 1, the trivial solution would be to set X = e. For general var(X) = v (which must be greater than or equal to 1,) use this formula for generating X:

 X = e(1:500) + sqrt(v-1)*randn(1,500);

where presumably 'e' and randn are statistically independent.

Answer 2

[Matt J]

How about just setting X=e ?

[John D'Errico]

Except the question was about the covariance. If the correlation was to be 1, then X=e would be correct.

[Roger Stafford]

Setting X = e does give a covariance of 1, because e was stated as N(0,1).

[John D'Errico]

Oops, yes. I missed that.

Answer 3

[the cyclist]

If you have the Statistics Toolbox, you can use the mvnrnd command.

If you don't, you can use Cholesky decomposition to do this. This page seems to have a pretty good explanation, and there is even some MATLAB code there.