Chapter 4 Discrete Random Variables

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Chapter 4 Discrete Random Variables

A random variable is a variable that assumes numerical values associated with the random outcomes
of an experiment, where one and only one numerical value is assigned to each sample point.
Example:

1. Toss a coin twice. The number of heads observed in a two tosses of a coin is a random
variable.

2. Roll a pair of fair dice once. The sum of numbers of dots on the upper faces of the two dice is
a random variable.

Section 4.1 Probability Distributions for Discrete Random Variables

The Probability distribution of a random variable, denoted by the symbol p(x), is a table, graph, or
formula that specifies all the values the random variable can assume and the probability associated
with each value.

Example 1: Consider the experiment of tossing a coin twice, and let x represent the number of heads
observed. Find the probability distribution of the random variable x.

Example 2: Roll a pair of fair dice once and let x equal the sum of numbers of dots on the upper faces
of the two dice. Give the probability distribution of x. What is the probability that the sum is less 5?

Example 3: The YSU bookstore has six special drafting pencils for sale, two of which are defective. A
student buys two of these pencils, selected at random. Let x be the number of defective pencils
purchased. Construct the probability distribution for x.

Probability distributions must follow some rules:

a. p(x) ≥ 0, for all values of x.
b. p(x) = 1, where the summation of p(x) is over all possible values of x.

Example : Explain why each of the following is or is not a valid probability distribution for a

discrete random variable x.

x1 2 5 x -3 4 5
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p(x) .2 .3 .45 p(x) .25 .15

Section 4.2 Expected Values of Discrete Random Variables

1. Mean
Suppose that all the possible values that a random variable x can assume are x1, x2, …, xk, and

that P(xi) is the associated probability of value xi. The mean, or expected value, of the random
variable is

= E(X) = xi p(xi )

Example: A teenager is a member of the caddy program at the Montebello Country Club. Each
morning he arrives at the golf course and enters his name into the caddy pool. The probability of
being selected on any day is 4/5, and if selected he will earn $50. On days when he is not selected,
he earns nothing. How much money does this caddy earn per day on average?

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Example: Rolling a die. Suppose that you pay $2.00 for each game. When outcome is 6, you win
$3.00. If outcome is 4 or 5, you win $1.00. If other outcomes occur, you lose your bet. How much
money do you expect to win per game? Interpret your result.

2. Variance and Standard Deviation
Suppose that all the possible outcomes in a sample space of a random experiment are x1, x2,

…, xk, and that p(xi) is the probability of outcome xi. The variance of a random variable x is
σ2 = [(x )2 ] = (xi )2 p(xi )
=(x1- )2P(x1) + (x2- )2 P(x2) + …+ (xk- )2 P(xk)

OR
= [xi2 p(xi )] 2

The standard deviation of a discrete random variable is equal to the square root of the variance.

Example: Consider the probability distribution for the random variable shown here:
x 10 12 18 20
p(x) .2 .3 .1 .4

a. Find μ and σ.

b. What is P(x < 15)?

c. Calculate μ ± 2σ.

d. What is the probability that x is in the interval μ ± 2σ? Compare your answer with
Chebyshev’s rule and the empirical rule.

Section 4.3 The Binomial Random Variable

1. Basic Concepts

A Bernoulli Trial is a random experiment that has only two outcomes or outcomes that can be
reduced to two outcomes. These outcomes are usually classified as successes S or failures F.

Example: 1). Toss a coin. Two outcomes: heads or tails.
2). Play a game. Win or lose.
3). Roll a die. Getting a 5 (success) or not getting a 5 (failure).

A binomial experiment is a sequence of n Bernoulli trials, and the number of successes in n trials is a
binomial random variable.

Characteristics of a Binomial Random Variable
a. The experiment consists of n identical Bernoulli trials.
b. The probability remains the same from trial to trial. Denote P(S) = p and P(F) = q = 1 – p.
c. The trials are independent.
d. The binomial r.v. x counts the number of successes in n trials.

Examples of binomial experiments and binomial r.v’s:
1). Toss a coin five times. The number of heads observed is a binomial r.v.
2). Play a game five times. The number of times of winning the game is a binomial r.v.
3). Roll a die twice. The number of times you observe a 3 is a binomial r.v.

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2. The Binomial Probability Distribution
Q: Roll a balanced die 5 times. What is the probability of getting exactly two 4’s?
Binomial Probability Formula

In a binomial experiment involving n independent and identical Bernoulli trials each with probability of
success p, the probability of having x successes is.

p( x) n p x qn x n! p x qn x , for x = 0, 1, .., n, where
x x!(n x)!

n The number of trials
x The number of successes in n trials
p The probability of a success
q The probability of a failure

Example: For the experiment of rolling a die 5 times, what is the probability of getting exactly
two 4’s?

Example: 75% of all cameras sold at New York Camera and Video are digital. The remaining
25% are traditional film cameras. Suppose 10 customers buying cameras at this store are selected
at random.

a. Find the probability that exactly 6 customers will buy digital cameras.

b. Find the probability that at least 8 customers will buy digital cameras.

3. Mean, Variance, and Standard Deviation for the Binomial Distribution

For a random variable that has the binomial distribution,

Mean of the distribution, = n·p
Variance of the distribution, 2 = n·p·q
Standard deviation, , is the square root of variance.
Example: Example 1 in Section 4.1.

Example: A die is rolled twice. Let x equal the number of 4’s observed.
1. Indentify the sample points associated with this experiment and assign a value of x to each
sample point.

2. Construct a probability distribution for x.
3. Find E(x).
4. Give the binomial probability distribution for x as a formula.
5. Use the formula for binomial probability distribution to construct a probability distribution for x.
6. Use the formula for binomial probability distribution to find E(x).

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