Chapter 4 Multivariate Normal Distribution

4.1 Random Vector Random Variable Random Vector X X 1, , X p are random variables

A. Cumulative Distribution Function (c.d.f.) Random Variable F(x) = P(X  x) F(x) = F(x 1, ,x p ) = P(X 1  x 1, , X p  x p ) Marginal distribution F(x 1 ) = P(X 1  x 1 ) = P(X 1  x 1, X 2 , , X p  ) = F(x 1, , ,  ) F(x 1, x 2 ) = P(X 1  x 1,X 2  x 2 ) = F(x 1, x 2, , ,  ) Random Vector

B. Density Random Variable Random Vector

C. Conditional Distribution Random Variable Random Vector Conditional Probability of A given B when A and B are not independent Conditional Density of x 1, , x q given x q+1 = x q+1, , x p = x p. h g where h: the joint density of x 1, , x p ; g: the marginal density of x q+1, , x p.

D. Independence Random Variable Random Vector (X 1,X 2 ) ~ F(x 1, x 2 ) If F(x 1, x 2 )= F 1 (x 1 ) F 2 (x 2 ),  x 1, x 2  x 1 and x 2 are said to be independent. (X 1, ,X p ) ~ F(x 1, ,x p ) If  X 1, ,X p are said to be mutually independent.

(X 1,X 2 ) ~ F(x 1, x 2 ) If F(x 1, x 2 )= F 1 (x 1 ) F 2 (x 2 ),  x 1, x 2  x 1 and x 2 are said to be independent. (X 1, ,X p ) ~ F(x 1, ,x p ) If  X 1, ,X p are said to be mutually independent. Random Vector X ~ F(x 1, ,x p ), Y ~ G(y 1, , y q ) X and Y are independent if

E. Expectation Random Variable Random Vector

Some Properties: E(AX) = AE(X) E(AXB + C) = AE(X)B + C E(AX + BY) = AE(X) + BE(Y) E(tr AX) = tr(AE(X))

F. Variance - Covariance Random Variable Random Vector

Other Properties:Cov(x) = Cov(x, x) Cov(Ax, By) = A Cov(x, y) B Cov(Ax) = A Cov(x) A Cov(x - a, y - b) = Cov(x, y), where a and b are constant vectors Cov(x - a) = Cov(x), where a is constant vector E(xx) = Cov(x) + E(x)E(x) E(x - a)(x - a) = Cov(x) + (E(x)- a)(E(x)- a)  a  R n Assume that E(x)=  and Cov(x) =  exist, and A is an p  p constant matrix, then E(xAx) = tr(A  ) +  A 

G. Correlation Random Variable Random Vector x = (X 1, ,X p ) that is called correlation matrix of x. Corr(x) = (Corr(X i,X j )): p  p

4.2 Multivariate Normal Distribution Random Variable: X ~ N( ,  2 )

Definition of Multivariate Normal Distribution standard normal:y = (Y 1, ,Y q ), Y 1, ,Y q i.i.d, N(0, 1) y ~ N q (0, I q )

Definition of Multivariate Normal Distribution

4.3 The bivariate normal distribution

The density function x is The contour of p( x 1, x 2 ) is an ellipsoid

4.4 Marginal and conditional distributions Theorem 4.4.1

Corollary 1 Corollary 2 All marginal distributions of are still normal distributions. Example Then,

The distribution of Ax is multivariate normal with mean And covariance matrix

Theorem Let x be a p × 1 random vector. Then x has a multivariate normal distribution if and only if a’x follows a normal distribution for any. Note:

Theorem The assumption is the same as in corollary 1 of Theorem Then the conditional distribution of x 1 given x 2 = x 2 is where Example 4.4.2

Example 1 Let x = ( x 1, …, x s ) be some body characteristics of women, where x 1 : Hight ( 身高 ) x 2 : Bust ( 胸圍 ) x 3 : Waist ( 腰圍 ) x 4 : Height below neck ( 頸下高度 ) x 5 : Buttocks ( 臀圍 )

The correlation of R can be computer from Take x (1) = ( x 1, x 2, x 3 ), x (1) = ( x 4 ) and x (3) = ( x 5 ).

Homework 3.5. Please directly computeand computer it by the recursion formula.

We see that

4.5 Independent Theorem Corollary 1