Chapter 3 Multivariate Random Variables

F(x1,x2,…xn)= P(X1≤x1,X2 ≤ x2,...,Xn ≤ xn) 3.1 Two-Dimensional Random Variables 1.n-dimensional variables n random variables X1，X2，...,Xn compose a n-dimensional vector (X1,X2,...,Xn), and the vector is named n-dimensional variables or random vector. 2. Joint distribution of random vector Define function F(x1,x2,…xn)= P(X1≤x1,X2 ≤ x2,...,Xn ≤ xn) the joint distribution function of random vector (X1,X2,...,Xn).

The Joint cdf for Two random variables Definition 3.1-P53 Let (X, Y) be 2-dimensional random variables. Define F(x,y)=P{Xx, Yy} the bivariate cdf of (X, Y) . Geometric interpretation：the value of F( x, y) assume the probability that the random points belong to area in dark

For (x1, y1), (x2, y2)R2, (x1< x2， y1<y2 ), then P{x1<X x2， y1<Yy2 } ＝F(x2, y2)－F(x1, y2)－ F (x2, y1)＋F (x1, y1). (x1, y2) (x2, y2) (x1, y1) (x2, y1)

EX Suppose that the joint cdf of (X,Y) is F(x,y), find the probability that (X,Y) stands in area G . G Answer

Joint distribution F(x, y) has the following characteristics: (1) For all (x, y) R2 , 0 F(x, y)  1,

(2) Monotonically increment for any fixed y R, x1<x2 yields F(x1, y)  F(x2 , y)； for any fixed x R, y1<y2 yields F(x, y1)  F(x , y2). (3) right continuous for xR, yR,

(4) for all (x1, y1), (x2, y2)R2, (x1< x2， y1<y2 ), F(x2, y2)－F(x1, y2)－ F (x2, y1)＋F (x1, y1)0. Conversely, any real-valued function satisfied the aforementioned 4 characteristics must be a joint distribution function of 2-dimensional variables.

Example 1. Let the joint distribution of (X,Y) is Find the value of A，B，C。 Find P{0<X<2,0<Y<3} Answer

P{X＝xi, Y＝ yj,}＝ pij ， (i, j＝1, 2, … ). Discrete joint distribution If both x and y are discrete random variable, then,(X, Y) take values in (xi, yj), (i, j＝1, 2, … ), it is said that X and Y have a discrete joint distribution . Definition 3.2-P54 The joint distribution is defined to be a function such that for any points (xi, yj), P{X＝xi, Y＝ yj,}＝ pij ， (i, j＝1, 2, … ). That is (X, Y)～ P{X＝xi, Y＝ yj,}＝ pij ，(i, j＝1, 2, … )，

The joint distribution can also be specified in the following table X Y y1 y2 … yj … p11 p12 ... P1j ... p21 p22 ... P2j ... pi1 pi2 ... Pij ... x1 x2 xi ... ... ... ... ... ... ... ... Characteristics of joint distribution : pij 0 , i, j＝1, 2, … ； (2)

Example 2. Suppose that there are two red balls and three white balls in a bag, catch one ball from the bag twice without put back, and define X and Y as follows: Please find the joint pmf of (X,Y) Y 1 0 X 1 0

Continuous joint distributions and density functions 1. It is said that two random variables (X, Y) have a continuous joint distribution if there exists a nonnegative function f (x, y) such that for all (x, y)R2，the distribution function satisfies and denote it with (X, Y)～ f (x, y)， (x, y)R2

2. characteristics of f(x, y) (1) f (x, y)0, (x, y)R2; (2) (3) (4) For any region G R2,

EX Let Find P{X>Y} y 1 G 1 x

EX 1 1 Find (1)the value of A； (2) the value of F(1,1)； (3) the probability of (X, Y) stand in region D：x0, y0, 2X+3y6 1 Answer (1) Since 1

(3)