Multivariate Distributions For discrete random vector pmf fxfx

Multivariate Distributions • For discrete random vector pmf: f(x)=f(x 1, . . . xn)=P(X 1=x 1, . . . Xn=xn) • For continuous random vector pdf is a function that satisfies

Multivariate Distributions • Expectations for discrete for continuous • Marginal Distributions for discrete for continuous

Multivariate Distributions • Conditonal Distribuitons The conditonal distribution of (Xk+1, . . . Xn) is defined by

Multivariate Distributions • Multinomial distribution with m trials and cell probabilities p 1, . . . pn • The conditonal distribution of any subset of multivariate distributions is still a multivariate distribution

Multivariate Distributions • Let X 1, . . . Xn be random vectors with joint pdf or pmf f(x 1, . . . xn), and f. Xi(xi) donotes the marginal pdf or pmf of Xi, then X 1, . . . Xn are called mutually independent random vectors if , for every (x 1, . . . xn) • Mutually independent is powerful than pairwise independent

Multivariate Distributions • Theorem 4. 6. 6 If X 1, . . . Xn are mutually independent Then • Theorem 4. 6. 7 If X 1, . . . Xn are mutually independent with mgfs Mx 1(t), Mxn(t). Then the mgf of Z=X 1+. . . Xn is

Multivariate Distributions • Theorem 4. 6. 11 X 1, . . . Xn are mutually independent if and only if f(x 1, . . . xn) can be written as f(x 1, . . . xn)=g 1(x 1). . . gn(xn) • Theorem 4. 6. 12 If X 1, . . . Xn are mutually independent , and gi(xi)is a function only of Xi, i=1, . . . , n. The Ui=gi(xi), i=1, . . . , n, are mutually independent

Multivariate Distributions • Transformation of multivariate pdf

Inequalities • Lemma 4. 7. 1 If a>0, b>0, p>1, q>1 , then with equality if and only if ap =bq • Theorem 4. 7. 2(Hölder's Ineauality) Let X and Y be any two random variables, and p, q saitisfy 4. 7. 1. Then

Inequalities • Theorem 4. 7. 3 (Cauchy-Schwarz Inequality) For any two random variables X and Y, • Theorem 4. 7. 5 (Minkowski's Inequality) For any two random variables X and Y, ,

Inequalities • A function g(x) is convex if for all x 1 , x 2 and. The function g(x)is concave if -g(x) is convex. • Theorem 4. 7. 7 (Jensen's Inequality) For any two random variable X, if g(x) is convex funciton, then

Inequalities • Theorem 4. 7. 7 (Jensen's Inequality) Equality holds if and only if , for every line a+bx that is tangent to g(x) at x=EX, p(g(X)=a+b. X)=1. • Theorem 4. 7. 9 (Covariance Inequality) Let X be any random variable and g(x) and h(x) any funcions such that Eg(X), Eh(X) exist.

Inequalities • Theorem 4. 7. 9 (Covariance Inequality) a. If g(x) is a nondecreasing function and h(x) is a nonincreasing function , then b. If g(x) and h(x) are either both nondecreasing function or nonincreasing function , then