# RE: st: Drawing 3 normally distributed variable with a given correlation structure to an existing variable

 From "Feiveson, Alan H. (JSC-SK311)" To Subject RE: st: Drawing 3 normally distributed variable with a given correlation structure to an existing variable Date Wed, 26 Sep 2007 12:36:31 -0500

```Rachel - Given the entire 4 x 4 covariance matrix and 4 x 1 mean vector,
you can show that the joint distribution of X2, X3 and X4 given X1 is
joint normal with mean vector

mu2 + S21(S11^-1)(X1 - mu1)

and covariance matrix

S22 - S21(S11^-1)S12

where mu2 is the 3 x 1 mean vector for (X2,X3, X4)' and mu1 = E(X1)
(scalar) and

where the entire covariance matrix is (S11 S12 \ S21 S22)

and where S11 is 1 x 1, S12 is 1 x 3, S21 is 3 x 1, and S22 is 3 x 3.

[for example, see Ch 2 in T. W. Anderson  - Introducton to Multivariate
Statistics (Wiley) ]

So given X1, you can then generate the other three using drawnorm and
the above  conditional mean and covariance matrix.

Al Feiveson

-----Original Message-----
From: owner-statalist@hsphsun2.harvard.edu
[mailto:owner-statalist@hsphsun2.harvard.edu] On Behalf Of Rachel
Sent: Wednesday, September 26, 2007 10:36 AM
To: statalist@hsphsun2.harvard.edu
Subject: Re: st: Drawing 3 normally distributed variable with a given
correlation structure to an existing variable

Hi Jeph,

-drawnorm- will draw 4 new variables.  In my case, X1~N(0,1) is already
given, and I'm trying to draw X2,X3,X4~N(0,1) such that the four
variables have a given correlation structure.
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```