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4.2 Probability Distributions for Discrete Random Variables

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 x of a discrete random variable X is the probability P(x) that X will take the value x in one trial of the experiment.

Definition

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

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

Each probability P(x) must be between 0 and 1: 0≤P(x)≤1.

The sum of all the probabilities is 1: ΣP(x)=1.

EXAMPLE 1

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

Construct the probability distribution of X.

Find the probability that at least one head is observed.

Solution:

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

xP(x)00.2510.5020.25

This table is the probability distribution of X.

“At least one head” is the event X ≥ 1, which is the union of the mutually exclusive events X = 1 and X = 2. Thus

P(X≥1)=P(1)+P(2)=0.50+0.25=0.75

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

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