Chapter 4 Multivariate Normal Distribution

1. Chapter 4Multivariate Normal Distribution

2. 4.1 Random Vector Random Variable X Random Vector X1, , Xp are random variables

3. A. Cumulative Distribution Function (c.d.f.) Random Variable F(x) = P(X  x) Random Vector F(x) = F(x1,,xp) = P(X1  x1, , Xp  xp) Marginal distribution • F(x1) = P(X1x1) = P(X1x1, X2, , Xp) = F(x1, , , ) • F(x1,x2) = P(X1x1 ,X2x2) = F(x1, x2, , , )

4. B. Density Random Variable Random Vector

5. C. Conditional Distribution Random Variable Conditional Probability of A given B when A and B are not independent Random Vector Conditional Density of x1,, xq given xq+1=xq+1, , xp= xp . h g where h: the joint density of x1,, xp; g: the marginal density of xq+1, , xp .

6. D. Independence Random Variable Random Vector (X1,,Xp) ~ F(x1, ,xp) If  X1, ,Xp are said to be mutually independent. (X1,X2) ~ F(x1, x2) If F(x1, x2)= F1 (x1) F2 (x2) ,  x1, x2  x1 and x2 are said to be independent.

7. Random Vector (X1,,Xp) ~ F(x1, ,xp) If  X1, ,Xp are said to be mutually independent. (X1,X2) ~ F(x1, x2) If F(x1, x2)= F1 (x1) F2 (x2) ,  x1, x2  x1 and x2 are said to be independent. X ~ F(x1, ,xp), Y ~ G(y1, , yq) X and Y are independent if

8. E. Expectation Random Variable Random Vector

9. 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))

10. F. Variance - Covariance Random Variable Random Vector

11. Other Properties: Cov(x) = Cov(x, x) Cov(Ax, By) = A Cov(x, y) B Cov(Ax) = A Cov(x) A Cov(x- a) = Cov(x) , where a is constant vector Cov(x- a, y- b) = Cov(x, y), where a and b are constant vectors E(xx) = Cov(x) + E(x)E(x) E(x - a)(x - a) = Cov(x) + (E(x)- a)(E(x)- a)  aRn Assume that E(x)=m and Cov(x) = S exist, and A is an pp constant matrix, then E(xAx) = tr(AS) + m Am

12. G. Correlation Random Variable Random Vector x = (X1, ,Xp) that is called correlation matrix of x . Corr(x) = (Corr(Xi,Xj)): pp

13. 4.2 Multivariate Normal Distribution Random Variable: X ~ N(m,s2)

14. Definition of Multivariate Normal Distribution standard normal:y = (Y1,,Yq), Y1,,Yq i.i.d, N(0, 1) y ~ Nq(0, Iq)

15. Definition of Multivariate Normal Distribution

16. 4.3 The bivariate normal distribution

17. The density functionx is The contour of p(x1, x2) is an ellipsoid

18. 4.4 Marginal and conditional distributions Theorem 4.4.1

19. Corollary 1 Corollary 2 All marginal distributions of are still normal distributions. Example 4.4.1 Then,

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

21. Theorem 4.4.2 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:

22. Theorem 4.4.3 The assumption is the same as in corollary 1 of Theorem 4.4.1. Then the conditional distribution of x1 given x2 = x2 is where Example 4.4.2

23. Example 1 • Let x = (x1, …, xs) be some body characteristics of women, where x1: Hight (身高) • x2: Bust (胸圍) • x3: Waist (腰圍) • x4: Height below neck (頸下高度) • x5: Buttocks (臀圍)

24. The correlation of R can be computer from Take x(1)= (x1, x2, x3), x(1)= (x4) and x(3)= (x5).

27. Homework 3.5. Please directly compute and computer it by the recursion formula.

28. We see that

29. 4.5 Independent Theorem 4.5.1 Corollary 1