# Random Variables and Probability Distributions

##### LEARNING OBJECTIVES

- To learn the concept of the probability distribution of a discrete random variable.
- To learn the concepts of the mean, variance, and standard deviation of a discrete random variable, and how to compute them.

##### Probability Distributions

Associated to each possible value of a discrete random variable is the probability that will take the value in one trial of the experiment.

##### Definition

The probability distribution of a discrete random variable is a list of each possible value of together with the probability that takes that value in one trial of the experiment.

The probabilities in the probability distribution of a random variable must satisfy the following two conditions:

1. Each probability must be between o and 1: .

2. The sum of all the probabilities is .

##### EXAMPLE 1

A fair coin is tossed twice. Let be the number of heads that are observed.

a. Construct the probability distribution of .

b. Find the probability that at least one head is observed.

Solution:

a. The possible values that can take are 0,1 , and 2 . Each of these numbers corresponds to an event in the sample space of equally likely outcomes for this experiment: to to , and to . The probability of each of these events, hence of the corresponding value of , can be found simply by counting, to give

0 | 1 | 2 | |

0.25 | 0.50 | 0.25 |

b. "At least one head" is the event , which is the union of the mutually exclusive events and . Thus

A histogram that graphically illustrates the probability distribution is given in Figure 4.1 "Probability Distribution for Tossing a Fair Coin Twice".

Figure 4.1

Probability Distribution for Tossing a Fair Coin Twice

##### EXAMPLE 2

A pair of fair dice is rolled. Let denote the sum of the number of dots on the top faces.

a. Construct the probability distribution of .

c. Find the probability that takes an even value.

Solution:

The sample space of equally likely outcomes is

11 | 12 | 13 | 14 | 15 | 16 |

21 | 22 | 23 | 24 | 25 | 26 |

31 | 32 | 33 | 34 | 35 | 36 |

41 | 42 | 43 | 44 | 45 | 46 |

51 | 52 | 53 | 54 | 55 | 56 |

61 | 62 | 63 | 64 | 65 | 66 |

a. The possible values for are the numbers 2 through 12 . is the event , so is the event , so . Continuing this way we obtain the table

This table is the probability distribution of .

b. The event is the union of the mutually exclusive events , and . Thus

c. Before we immediately jump to the conclusion that the probability that takes an even value must be , note that takes six different even values but only five different odd values. We compute

A histogram that graphically illustrates the probability distribution is given in Figure 4.2

"Probability Distribution for Tossing Two Fair Dice".

Figure 4.2

Probability Distribution for Tossing Two Fair Dice