Hi guys, I have two randomly generated variables and want to generate the third which is correlated with one of them and uncorrelated with the other. How can I generate such random variable?

To be specific I have these series

e~N(0,1)and x~N(0,v), where v is variance of x

and I want to generate another variable say Y that satisfy the following conditions

E(y,e) equal to zero E(Y,x) not equal to zero

Please help me out with this.

[Harry]

Does your notation "E(a,b)" basically mean the correlation/covariance between a and b? (It doesn't matter which).

When you say "E(y,e) equal to zero", do you mean "E(Y,e) equal to zero"?

Do we assume that e and x are uncorrelated? This would make things easier in my opinion.

[Solomon Gofere]

Sorry I was not clear with my question. My notations are like this: E(x,y) refers to correlation between the two. Y and y are exactly the same, i just missed to hit shift. And e and x are correlated.

Thanks

[Harry]

That's good - the answer I wrote should work for you. Please accept it if you are happy with it.

Answer 1

[Harry]

With problems like this, I always think it is easiest to start with zero-mean, unit-power uncorrelated random variables. It's difficult to write equations in this forum, so I wrote some which I have uploaded as an image at the end of this post. I assume you want real numbers, but this is straightforward to extend to the complex case.

First, here's the Matlab code:

close all; clear all; clc;

% Draw N random values for e
N = 10000;
e = randn(N,1);

% Set required variances and cross-correlation for x and y
Px = 1;
Py = 3;
rho_xy = 0.4;

% Generate s1 and s2 of length N
s1 = randn(N,1);
s2 = randn(N,1);

% Make x
x = sqrt(Px)*s1;

% Make y
B = rho_xy*sqrt(Py);
C = sqrt(Py*(1 - rho_xy^2));
y = B*x + C*s2;

% Check results
sample_Pe = mean(e.^2)
sample_Px = mean(x.^2)
sample_Py = mean(y.^2)
sample_rho_ey = e.'*y/(norm(e)*norm(y))
sample_rho_xy = x.'*y/(norm(x)*norm(y))

And here are the equations: