Understanding Two-Dimensional Random Variables and Their Joint Distributions

1. Chapter 3 Multivariate Random Variables

2. 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).

3. 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

4. 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)

5. 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

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

7. (2) Monotonically increment for any fixed y R, x1<x2yields F(x1, y)  F(x2 , y)； for any fixed x R, y1<y2yields F(x, y1)  F(x , y2). (3) right continuous for xR, yR,

8. (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.

9. 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

10. 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 Thejoint 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, … )，

11. The joint distribution can also be specified in the following table X Y y1 y2 … yj … p11p12 ...P1j ... p21p22 ...P2j ... pi1pi2 ...Pij ... x1 x2 xi ... ... ... ... ... ... ... ... • Characteristics of joint distribution : • pij0 , i, j＝1, 2, … ； • (2)

12. 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

13. 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

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

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

16. EX 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

17. (3)