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10.2 The Multivariate Normal Distribution

With the matrix notation set up the bivariate normal distribution is easily extended to higher dimensions. The vector 𝑿=(X1,,Xd) is said to have a d-dimensional normal distribution if its pdf is given for all 𝒙 by

f𝑿(𝒙)=1(2π)d/2detΣexp{-12(𝒙-𝝁)Σ-1(𝒙-𝝁)},

where 𝝁=(μ1,,μd) is the vector of expectations and Σ is the variance-covariance matrix with (i,j)-th element σij being the pairwise covariance of variables Xi and Xj. As

ρij=𝖢𝗈𝗋𝗋[Xi,Xj]=𝖢𝗈𝗏[Xi,Xj]𝖵𝖺𝗋[Xi]𝖵𝖺𝗋[Xj]=Σijσiσj,

the variance-covariance matrix can be written

Σ=[Σ11Σ1dΣijΣjiΣd1Σdd]=[σ12σ1σdρ1dσiσjρijσiσjρjiσ1σdρd1σd2],

where σi2 is the variance of Xi, and ρij is the correlation of Xi and Xj.

We often denote the distribution of 𝑿 by

𝑿𝖬𝖵𝖭d(𝝁,Σ),

where 𝖬𝖵𝖭d stands for multivariate normal distribution of d dimensions.

Proposition 10.1.1 showed that a multivariate normal vector can always be written as the sum of an expectation vector, 𝝁, and a linear combination of iid 𝖭(0,1) random variables.

Therefore as in the bivariate case the univariate marginal distributions are all normal

XiN(μi,σi2),

for i=1,,d. In fact, any linear combination of the X’s will again be normal

i=1daiXiN(i=1daiμi,i=1dj=1daiajΣij),

where the expressions for the expectation and variance follow from Chapter 8; the fact that the distribution is normal is the convolution property of the normal distribution. Using that σii=σi2 and σij=σji the variance can also be written as

i=1dai2σi2+2i=1dj=1i-1aiajΣij=i=1dai2𝖵𝖺𝗋[Xi]+2i=1dj=1i-1aiaj𝖢𝗈𝗏[Xi,Xj].

The bivariate marginal distribution of any pair of variables (Xi,Xj) is also normal with expectation and variance given by,

[XiXj]𝖬𝖵𝖭2([μiμj],[σi2ρijσiσjρijσiσjσj2]).

In general all the lower order marginal distributions, i.e. the distribution of any subset of the X’s, will be normal with expectation vector and variance matrix given by deleting the rows and columns corresponding to the variables we leave out.

Indeed, any multi-dimensional linear transformation of 𝑿 will be normal

A𝑿𝖬𝖵𝖭m(A𝝁,AΣA),

where A is an m×d matrix of constants.

It also turns out that all the conditional distributions of a subset of the X’s given the rest is again normal.

Example 10.2.1.

Let (X1,X2,X3,X4,X5) be multivariate normal with

[X1X2X3X4X5]𝖬𝖵𝖭5([01-204],[261515-7-11518-2013015-2099-440-713-4414373-1007390]).

Find the marginal distribution of (X1,X3,X4).

Solution.  The marginal distribution is obtained by deleting rows 2 and 5 of the expectation vector, and rows 2 and 5 and columns 2 and 5 of the variance matrix. Thus

[X1X3X4]𝖬𝖵𝖭3([0-20],[2615-71599-44-7-44143]).

The most important special case is when X1,,Xd are i.i.d. N(μ,σ2) random variables. In this case

𝝁=[μμ],

and

Σ=[σ2000000σ2]=σ2Id,

where Id is the identity matrix in d dimensions, i.e. a d×d matrix with ones in the diagonal and zeros everywhere else. The pdf becomes

f𝑿(𝒙)=1(2π)d/2σdexp{-12σ2i=1d(xi-μ)2},

since Σ-1=1σ2Id.