1. Chapter 3 Multivariate Random Variables

2. 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).

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 ,  F(x, y)  1,

7. (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,

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. 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, … )，

11. The joint distribution can also be specified in the following table
X Y y1 y2 … yj …
p p P1j
p p P2j
pi pi 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 X

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 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

17. (3)