# How to generate 'random' values of a variable known to be correlated with 5 other variables?

I do individual-based simulations and I am trying to initialize a variable P known to be correlated to another five variables: H(r=-0.67), E(r=-0.33), X(r=0.33), A(r=-0.25), C(r=-0.25), O(r=0.25). These five variables are assumed to be independent and the value of each these variables is drawn from a normal distribution with mean=0.5 and SD=0.25. My question is if there is a way in which I can generate 'random' values of P which respect the correlations with each of the other five variables H,E,X,A,C,O?

I am not sure this is even possible, any help will be greatly appreciated.

## 1 Answer

Let S be the covariance matrix of random vector (P,H,E,X,A,C,O). Let

S = L*L'

be the Cholesky decomposition of matrix S. Simulate 7 standard normal variables: Z1, Z2, Z3, Z4, Z5, Z6, Z7. Then

(P,H,E,X,A,C,O) = L * (Z1, Z2, Z3, Z4, Z5, Z6, Z7) + 0.5.

I assume the expectation of P is 0.5 as well.

• Thanks for your answer. I will look into it although the thing with the P variable is a bit more complicated than I described. The value of P ranges between [0,1] with 98.75% of values between [0,0.25]; 1% between [0.25,0.5] and 0.25% [0.5,1]. Any thoughts on this would be greatly appreciated. – Ivan Puga Jan 17 '18 at 11:54