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