Def. Let x and y be two discrete random variables defined on the sample space of an experiment. The joint probability mass functions p(x,y) is defined for each pair of numbers (x,y) by:
p (x,y) = p (X = x, and Y = y) = fxy (x,y)
Let A be any set consisting of pairs of (x,y) values. Then the probability P [(x, y) A] is obtained by summing the joint pmf over pairs in A:
p [(x,y) A] =
Note A joint pmf must satisfy the following conditions.
p (x,y) 0
(Row sum or column sum = 1)
Ex. 1 In an automobile plant, two tasks are performed by robots. The first entails welding two joints; the second, tightening three bolts. Let X denote the number of defective welds and Y the number of improperly tightened bolts produced per car. Past data indicates that the joint probability density for (X,Y) is as shown below.