Joint Probability Distribution

Joint Probability Distribution

The joint probability distribution of two discrete random variables X and Y is a function whose domain is the set of ordered pairs (x, y) , where x and y are possible values for X and Y, respectively, and whose range is the set of probability values corresponding to the ordered pairs in its domain. This is denoted by p_X,Y(x, y) and is defined as

pX,Y(x, y) = P (X = x and Y = y)

The definition of the joint probability distribution can be extended to three or more random variables. In general, the joint probability distribution of the set of discrete random variables X₁ , X₂, .... , X_n is given by

p ( x₁ , x₂ , .... , x_n ) = P (X₁ = x₁ and X₂ = x₂ and .... X_n = x_n )

EX. A box has 10 cartons. Two of them contain check prizes, three of them have gift certificates, and the rest are empty. Two cartons will be picked at random from the box.

Let the random variable X be the number of cartons with check prizes drawn, and let the random variable Y be the number of cartons with gift certificates drawn. To find the joint probability distribution of X and Y, we note that there are

where x is the number of cartons with check prizes selected, y is the number of cartons with gift certificates chosen, and (2 - x - y) is the number of empty cartons picked. Therefore, the joint probability distribution of X and Y is given by

for x Î { 0 , 1 , 2 } , y Î { 0 , 1 , 2 } , x + y £ 2

The above joint probability distribution of X and Y is tabulated as follows:

(x, y)	pX,Y (x, y)
Â	Â
(0, 0)	2/9
(0, 1)	1/3
(0, 2)	1/15
(1, 0)	2/9
(1, 1)	2/15
(2, 0)	1/45

Note that 2/9 + 1/3 + 1/15 + 2/9 + 2/15 + 1/45 = 1.

The joint probability function of two discrete random variables is related functionally to the probability mass function of either random variable. The probability mass function of a random variable can be derived from its joint probability distribution with another random variable (or a set of random variables) by summing the joint probability distribution across all possible values of the other random variable(s). In other words,

The probability mass functions p_X(x) and p_Y(y) are also known as the marginal distributions of X and Y, respectively.

In general, the probability mass function of a random variable X₁ can be derived from the joint probability distribution of the set of discrete random variables X₁ , X₂, .... , X_n .

Two random variables are independent if and only if their joint probability distribution function is simply the product of the simple probability distribution for each random variable. That is, the random variables X and Y are independent, if and only if

p_X,Y(x, y) = p_X(x) p_Y(y)