3.4. Jointly distributed discrete random variable

Although the probability distributions studied so far have involved only one random variable, many decisions are based upon an analysis of two or more random variables. In problem situations that involve two or more random variables, the resulting probability distribution is referred to as a joint probability distribution.

Example:

The number of between-meal snacks eaten by students in a day during final examinations week depends on the number of tests a student had to take on that day. The accompanying table shows joint probabilities, estimated from a survey.

Table 3.8

Number of snacks (Y)	Number of tests (X) 0 1 2	P(y)
0 1 2 3	0.05 0.08 0.09 0.07 0.09 0.11 0.11 0.04 0.10 0.08 0.07 0.11	0.22 0.27 0.25 0.26
P(x)	0.31 0.28 0.41	1.00

Definition:

Let X and Y be a pair of discrete random variables. Their joint probability function expresses the probability that simultaneously X takes the specific value x and Y takes the value y, as a function of x and y.

The notation used is so,

For example, . It means that, the probability that randomly chosen student has 2 tests and eats 3 snacks is 0.11.

Definition:

Let X and Y be a pair of jointly distributed random variables. The probability function of the random variable X is called its marginal probability function, denoted by , and is obtained by summing the joint probabilities over all possible values; that is

.

Similarly, the marginal probability function of the random variable Y is

.

Marginal probability functions and are shown in the lower row and the right column of the table 3.8.

For example, expresses the probability that, randomly chosen student has no tests is 0.31.

, expresses the probability that randomly chosen student eats 2 snacks is 0.25.

Properties of joint probability functions of discrete random variables

Let X and Y be a discrete random variables with joint probability function .Then

1. for any pairs of x and y.

2. The sum of the joint probabilities over all possible pairs

of values must be 1.

3.4.1. Conditional probability function

Let X and Y be a pair of jointly distributed discrete random variables. The conditional probability function of the random variable Y, given that the random variable X takes the value x, expresses the probability that Y takes the value y, as a function of y, when the value x is specified for X.

This is denoted by , and defined as

Similarly, the conditional probability function of X, given Y=y is

.

For example, using the table 3.8, we can compute the conditional probability of y=2, given that x=1 as

It means, the probability that randomly chosen student who has 1 test eats 2 snacks is 1/7.

The probability of , given that is

It means, the probability that randomly chosen student who eats 3 snacks has 2 tests is 11/26.