Joint Probability

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 S_X = {0, 1, 2, 3}so that random variables X and Y are discrete. and S_Y = {0,1}

The joint sample space is the two dimensional grid S_XY = {(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.