Suppose I wish to generate normals $X, Y, Z$ with the correlation matrix R but with means $0, 1, 2$, and variances $4, 16, 25$, respectively.
How would you do this?
The only way I know of doing this is to use a Cholesky Decomposition of a correlation matrix. Can I find a correlation matrix using the means and variances?
I plan to generate correlated standard normals using the typical procedure of Cholesky decomposing the correlation matrix and then multiply that new matrix L by a 1x3 vector of i.i.d standard normals(U,V,W). I would use the new correlated standard normals and shift and scale them as follows:
$Z_1=2*U_1+0$
$Y_1=4*V_1+1$
$X_1=5*W_1+2$
The transformations that you plan to apply to $U_1$, $V_1$, $W_1$ will result in variables $Z_1$, $Y_1$, $X_1$ with the same correlation structure compared to before, so your algorithm will have the desired effect (assuming the desire is to retain the correlation matrix as $R$). This follows from the definition of correlation $$ \operatorname{Corr}(X,Y)={\operatorname{Cov}(X,Y)\over\sqrt{\operatorname{Var}(X)}\sqrt{\operatorname{Var}(Y)}} $$ and the properties of covariance and variance: $$ \operatorname{Cov}(aX+b,cY+d)=ac\operatorname{Cov}(X,Y), $$ $$ \operatorname{Var}(aX+b)=a^2\operatorname{Var}(X). $$