ABSTRACT
Let X{, i = 1, be k independent normally distributed variables with means fii and variances af respectively, Xi ~ N [p i,a f]. Define a new variable Z to be the weighted sum Z = Y i aiX i. Then Z is also normally distributed. The mean and variance of Z are weighted sums of the means and variances respectively of the component X i :
k k
i=1 z=l k k
More generally, let the Xi variables be correlated with pairwise covari ances C [X i ,X j] = (Tij for i ^ j . Z is still normally distributed with the mean given previously; the variance, however, is now extended to include contributions from the covariances among the X i ,
k k i-1 v[z] = Z a> V[Xi] + 2 ajCl X^Xj]
2 = 1 2 = 1 j = 1 k k 2-1
= E a a^i + 2 E E aiai aii ' 2 = 1 2 = 1 j= l
A random p-vector X is said to be (jointly) normally distributed with mean vector [i and covariance matrix E if it has the (joint) probability density function
An equivalent definition is: a set of p random quantities Xi is jointly nor mally distributed if and only if every linear combination, XX=i ai^ i at least one az-^ 0, is normally distributed. We use the notation X ~ N[p, E].
