Post on 18-Jan-2016
Chapter 3 Multivariate 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,X
2,...,Xn).
3.1 Two-Dimensional Random Variables
Definition 3.1-P53
Let (X, Y) be 2-dimensional random variables. Define F(x,y)=P{Xx, Yy}
the bivariate cdf of (X, Y) .
The Joint cdf for Two random variables
00 ,,, yyxxyx
Geometric interpretation : t
he value of F( x, y) assume th
e probability that the random p
oints belong to area in dark
For (x1, y1), (x2, y2)R2, (x1< x2 , y1<y2 ), then
P{x1<X x2 , y1<Yy2 }
= F(x2, y2) - F(x1, y2) - F (x2, y1) + F (x1, y1).
(x1, y1)
(x2, y2)
(x2, y1)
(x1, y2)
1x 2x 3x
1y
2y
3ySuppose that the joint cdf of (X,Y) is F(x,y), find the probability that (X,
Y) stands in area G .
Answer
2 1 3 3 2 3 3 1
1 2 2 3 1 3 2 2
{( , ) } [ ( , ) ( , ) ( , ) ( , )]
[ ( , ) ( , ) ( , ) ( , )]
P X Y G F x y F x y F x y F x y
F x y F x y F x y F x y
L
Joint distribution F(x, y) has the following characteristics:
0),(lim),(
yxFFyx
1),(lim),(
yxFFyx
0),(lim),(
yxFyFx
0),(lim),(
yxFxFy
(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).
);y,x(F)y,x(Flim)y,0x(F 0xx
00
).y,x(F)y,x(Flim)0y,x(F 0yy
00
(3) right continuous for xR, yR,
(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
)]3
()][2
([),(y
arctgCx
arctgBAyxF
1) Find the value of A , B , C 。2) Find P{0<X<2,0<Y<3}
Answer 1]2
][2
[),(
CBAF
0)]3
(][2
[),( y
arctgCBAyF
0]2
)][2
([),(
Cx
arctgBAxF
2
1
2
ACB
16
1)0,2()3,0()3,2()0,0(}30,20{ FFFFYXP
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 t
hat X and Y have a discrete joint distribution .
Definition 3.2-P54
The joint distribution is defined to be a function such th
at 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, … ) ,
X Y y
1 y
2 … y
j …
p11 p12 ... P1j ...
p21 p22 ... P2j ...
pi1 pi2 ... Pij ...
......
...
...
...
...
... ...
Characteristics of joint distribution :(1) pij 0 , i, j = 1, 2, … ; (2) 1
1 1
= i j
ijp
x1
x2
xi
The joint distribution can also be specified in the following table
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:
ball whiteisput second the0
ball red isput second the1
ball whiteisput first the0
ball red isput first the1
Y
XPlease find the joint pmf of (X,Y)
XY
1 0
1 0
10
110
3
10
3
10
3
25
22}1,1{
P
PYXP
25
32}0,1{
PYXP
25
23}1,0{
PYXP
25
23}0,0{
P
PYXP
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)
);,(),(2
yxfyx
yxF
(3)
( , ) 1;f x y dxdy
- -
(4) For any region G R2,
G
dxdyyxfGYXP .),(}),{(
G
dxdyyxfGYXP .),(}),{(
Let
others
yxyxfYX
0
10,101),(~),(
Find P{X>Y}
2
11}{
0
1
0
x
dydxYXP
1
1
x
y
Find (1)the value of A ; (2) the value of F(1,1) ; (3) the probability of (X, Y) stand in
region D : x0, y0, 2X+3y6
casesother for ,0
0,0,),(~),(
)32( yxAeyxfYX
yx
Answer (1) Since
6 A
1
0
1
0
32)32( )1)(1(6)1,1()2( eedxdyeF yx
1
1
- -
-(2x+3y)
0 0
f(x, y)dxdy = Ae dxdy = 1
(3)
3
0
322
0
)32(6 dyedx
x
yx
671 e
dxdyeDYXPD
yx )32(6}),{(