Joint
Probability Distribution
Bi-variate
Distributions
Let X and
Y be discrete random variables with sample spaces Sx and Sy, respectively. The
likelihood that X and Y takes values in the joint sample space are given is determined
by the joint probability distribution p(x,y) = Pr ( X= x, Y=y). The function p(x,y)
satisfies:
(i)
p (x,
y) > 0 for x, y
(ii)
p(x, y) = 0, for x, y
(iii)
An example of Bi-variate discrete
distribution is as follows:
Let X denote the monthly return
in percent on Microsoft Stock, and let Y be the monthly return on Apple Stock. For
simplicity we suppose that the sample spaces for X and Y are SX
= {0, 1, 2, 3}so that random variables X and Y are discrete. and SY
= {0,1}
The joint sample space is the two dimensional
grid SXY = {(0,0), (0,1), (1,0), (1,0), (1,1), (2,0), (2,1),
(3,0), (3,1)}
The same is displayed in the
table below
|
%
|
Y
0 1
|
Pr (X)
|
|
0
X 1
2
3
|
1/8 0
2/8 1/8
1/8 2/8
0 1/8
|
1/8
3/8
3/8
1/8
|
|
|
4/8 4/8
|
1
|
As can be seen from the
table, the sum of all the entries in the table is unity.
