# Thread: Two Variables covariance matrix

1. ## Two Variables covariance matrix

Hello this is my first to your nice community

I know that a variable X that is assumed to follow the gaussian distribution is denoted like this
X~(median,variance).

I have found today the following
Yfrog Image : yfrog.com/b9correlationcoefficientsg
probably the author of the equation above is talking about two variables P and S. I do not know how to
a) read the equation above and
b) how to convert the variance correlation matrix to variance values for each of the two variables.

Best Regards
Alex

2. As much as it can be understood the author is either talking about two variables P and S or two vector variables.
In case of the first,
The interpretation is :
variables P and S are not independent. They are normally distributed with
E(P)=mu1
E(S)=mu2
Cov(P,S)= -rho.sigma1.sigma2
Var(P)=(sigma1)^2
Var(S)=(sigma2)^2

3. Originally Posted by Dooti
As much as it can be understood the author is either talking about two variables P and S or two vector variables.
In case of the first,
The interpretation is :
variables P and S are not independent. They are normally distributed with
E(P)=mu1
E(S)=mu2
Cov(P,S)= -rho.sigma1.sigma2
Var(P)=(sigma1)^2
Var(S)=(sigma2)^2
Thanks! Yeah! Actually it is the first case. But could you please let me know how did you calculate Cov(P,S). Do you also any external link to read about how to calculate such things. Best Regards

4. Originally Posted by dervast
Thanks! Yeah! Actually it is the first case. But could you please let me know how did you calculate Cov(P,S). Do you also any external link to read about how to calculate such things. Best Regards
The variance of a random vector $\displaystyle x$ is defined to be

$\displaystyle E\left( (x - \mu) (x - \mu)^T \right)$.

If you write $\displaystyle x = (x_1, x_2, ..., x_n)^T$, what this gives you is a matrix whose ij'th element is $\displaystyle \mbox{Cov}(x_i, x_j)$. So, in your example, if you want $\displaystyle \mbox{Cov}(P, S)$, all you do is look at the off diagonal elements of the matrix they give.

The notation they are using provides the parameters for a multivariate normal distribution.

5. Its a multivariate normal setup..quite common form. Actually the variance covariance matrix gives you all you need to know. The diagonal elements corresponds to individual variances of the variables and the off diagonal their respective correlations. That's all. You can look it uo in any book concerning the subject.