Section 6.1 – Discrete Random Variables

Objectives:

1. Distinguish between discrete and continuous random variables

2. Identify discrete probability distributions

3. Construct probability histograms

4. Compute and interpret the mean of a discrete random variable

5. Compute and Interpret the Expected Value of a Discrete Random Variable

6. Compute the standard deviation of a discrete random variable

Objective – Distinguish between Discrete and Continuous Random Variables

- – a numerical measure of the outcome from a probability

experiment, so its value is determined by chance.

o Random variables are denoted using letters such as x

- – either a finite or countable number of values

- – infinitely many values and are measureable

o The values of a continuous random variable can be plotted on a

line in an fashion

Math 203 Notes – Chapter 6 - Discrete Probability Distributions

Example: Distinguishing Between Discrete and Continuous Random Variables

Determine whether the following random variables are discrete or continuous. State possible values for the

random variable.

a) The number of light bulbs that burn out in a room of 10 light bulbs in the next year.

b) The number of leaves on a randomly selected oak tree.

c) The length of time between calls to 911.

Objective – Identify Discrete Probability Distributions

- – provides the possible values of the random variable x and

their

The table to the right shows the probability distribution for the random

variable x, where x represents the number of movies streamed on Netflix

each month.

Rules for a Discrete Probability Distribution

Let P(x) denote the probability that the random variable X equals x; then

1. Σ P(x) = 1

2. 0 ≤ P(x) ≤ 1

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Example: Identifying Probability Distributions

Are the following a probability distributions? If not, state why.

Example: Constructing Probability Distributions

The following data represent the number of games played in each World Series from 1923 to 2010.

Construct a probability distribution for the number of games played in a World Series.

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Objective – Construct Probability Histograms

- – similar to relative frequency histogram but

displaying probabilities instead

o horizontal axis corresponds to the of the random variable

o vertical axis represents the of that value of the variable occurring

Objective – Compute and Interpret the Mean and Expected Value of a Discrete Random Variable

The Mean of a Discrete Random Variable

The mean of a discrete random variable is given by the formula

where x is the value of the random variable and P(x) is the probability of observing the value x.

Because the mean of a discrete random variable represents what we would expect to happen in the long

run, it is also called the expected value and is denoted E(x),

! ! = !! = ! ⋅ ! !

µx = x ⋅P x( )"# $%∑

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Example: Computing the Mean of a Discrete Random Variable Compute and interpret the mean of the probability distribution, where x represents the number of movies

streamed on Netflix each month.

Interpretation of the Mean of a Discrete Random Variable

Suppose an experiment is repeated n independent times and the value of the random variable x is recorded.

As the number of repetitions of the experiment , the mean value of the n trials

(sample mean) will approach µx, the mean of the random variable X.

In other words, the difference between x and µx gets closer to 0 as

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Example: Computing the Expected Value of a Discrete Random Variable

A term life insurance policy will pay a beneficiary a certain sum of money upon the death of the

policyholder. These policies have premiums that must be paid annually. Suppose a life insurance

company sells a $250,000 one-year term life insurance policy to a 49-year-old female for $530. The

probability the female will survive the year is 0.99791. Compute the expected value of this policy to the

insurance company. Source: National Vital Statistics Report, Vol. 47, No. 28

Example: Computing the Expected Value of a Discrete Random Variable

Jack and Jill purchased a foreclosed property on a hill for $350,000 and spent an additional $35,000 fixing

it up. They feel there is 15% chance to resell the property for $520,000, a 35% chance to resell for

$475,000, and a 50% chance to resell for $415,000. Compute and interpret the expected profit for reselling

the property.

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Objective – Compute the Standard Deviation of a Discrete Random Variable

Standard Deviation and Variance of a Discrete Random Variable

The standard deviation of a discrete random variable X is given by

!! = ∑ !! ⋅ ! ! − !!!

where x is the value of the random variable, µx is the mean of the random variable, and P(x) is the

probability of observing a value of the random variable.

*Note, the variance of the variable is simply !!!

Example: Computing the Variance and Standard Deviation of a

Discrete Random Variable

Compute the variance and standard deviation of the following probability

distribution, where x represents the number of movies streamed on Netflix each

month.

*Remember, to attain variance, we simply compute

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Example: Computing the Mean and Standard Deviation of a Discrete Random Variable

From a previous example, we were given the following data representing the number of games played in each World Series from 1923 to 2010. We constructed a probability distribution for the number of games played in a World Series.

a) Compute and interpret the expected number of games played in a World Series

b) Compute the standard deviation of the number of games played in a World Series

We expect approximately 5.782 games to be played in the World Series

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We can also use our calculators to find the mean and standard deviation of a probability distribution:

From a previous example: Compute the mean and standard deviation of the probability

distribution to the right, where x represents the number of movies streamed on Netflix

each month.

Input the data x values into L1 and the probabilities into L2 then execute

1-Var Stats L1,L2

(Hit Stat, Calc, 1-Var Stats L1,L2, enter)

The mean of the distribution is the value of !.

The standard deviation of the distribution is the value of !! because

the probabilities given represent the probabilities for the entire

population.

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HUGEDEAL

Section 6.2 – The Binomial Probability Distribution

Objectives

1. Determine whether a probability experiment is a binomial experiment

2. Compute probabilities of binomial experiments

3. Compute the mean and standard deviation of a binomial random variable

4. Construct binomial probability histograms

Criteria for a Binomial Probability Experiment

An experiment is said to be a binomial experiment if

• The experiment is performed a number of times. Each repetition of

the experiment is called a trial.

• The trials are . This means the outcome of one trial

will not affect the outcome of the other trials.

• For each trial, there are two mutually exclusive (or disjoint) outcomes, success or failure.

• The probability of success is for each trial of the experiment.

Notation Used in the Binomial Probability Distribution

• There are n independent trials of the experiment.

• denotes the probability of success so that is the probability of failure

• x is a binomial random variable that denotes the number of successes in n independent trials of the

experiment. So, 0 < x < n

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Needtodothisbyhandandoncalculator!

Example: Identifying Binomial Experiments

Determine whether the following are binomial experiments

1) A player rolls a pair of fair die 10 times. The number, x, of 7’s rolled is recorded.

2) The 11 largest airlines had an on-time percentage of 84.7% in November, 2001 according to the Air

Travel Consumer Report. In order to assess reasons for delays, an official with the FAA randomly

selects flights until she finds 10 that were not on time. The number of flights, x, that need to be selected

until 10 were not on time is recorded.

3) In a class of 30 students, 55% are female. The instructor randomly selects 4 students. The number, x,

of females selected is recorded.

Objective – Compute Probabilities of Binomial Experiments

Binomial Probability Distribution Function

The probability of obtaining x successes in n independent trials of a binomial experiment is given by

where p is the probability of success.

First, some mathematical definitions: Symbol

“at least” or “no less than” or “greater than or equal to” ≥

“more than” or “greater than” >

“fewer than” or “less than” <

“no more than” or “at most” or “less than or equal to ≤

“exactly” or “equals” or “is” =

P x( ) = nCx px 1− p( )n−x x = 0,1, 2,...,n

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Example: Finding Probability of a Binomial Probability Distribution

1) According to the Air Travel Consumer Report, the 11 largest air carriers had an on-time percentage

of 79.0%. Suppose that 4 flights are randomly selected in May, and the number of on-time flights x

is recorded. Find the probability that exactly 2 flights are on time by hand.

2) According to the Experian Automotive, 35% of all car-owning households have three or more cars.

a. In a random sample of 20 car-owning households, what is the probability that exactly 5 have

three or more cars?

By Hand:

We can also use the binompdf operation on our calculators:

2nd, Dist (Vars), arrow down to binompdf(, input

binompdf(20,.35,5) ≈

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Binompdf versus Binomcdf commands

- Binompdf is used to find the probability of a

of successes

- Binomcdf is used to find the probability of

Continuing with the previous example, according to the Experian Automotive, 35% of all car-owning

households have three or more cars.

b. In a random sample of 20 car-owning households, what is the probability that less than 4

households have three or more cars?

By hand: We would have to find the probability of each number of successes occurring using the formula

then add the results since the events are disjoint. That is, by hand we would need to find

P(X < 4) = P(X ≤ 3) = P(0)+ P(1)+ P(2)+ P(3)

Using our calculators, we can use the binomcdf function: 2nd, Dist (Vars), arrow down to binomcdf(

c. In a random sample of 20 car-owning households, what is the probability that at least 4

households three or more cars? Careful using your calculator with these!

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Example: Finding Probability of a Binomial Probability Distribution

According to a study done by Nick Wilson of Otago University Wellington, the probability a randomly selected individual will not cover his or her mouth when sneezing is 0.267. Suppose you sit on a bench in a mall and observe people’s habits as they sneeze.

1) Find the probability, by hand, that among 10 randomly observed individuals exactly 4 do not cover their mouth when sneezing?

2) What is the probability that among 10 randomly observed individuals fewer than 3 do not cover their mouth?

3) Would you be surprised if, after observing 10 individuals, fewer than half covered their mouth when sneezing? Why?

4) What is the probability that among 10 randomly observed individuals 7 or more do not cover their mouth?

5) What is the probability that among 10 randomly observed individuals between 4 and 6 do not cover their mouth?

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Weneedtoknowthisbyhand!

Objective – Compute the Mean and Standard Deviation of a Binomial Random Variable

Mean (Expected Value) and Standard Deviation of a Binomial Random Variable

A binomial experiment with independent trials and probability of success has a

mean and standard deviation given by the formulas

The mean of a binomial random variable is interpreted as the expected number of successes in n trials of

the experiment.

Example: Finding the Mean and Standard Deviation of a Binomial Random Variable

According to the Experian Automotive, 35% of all car-owning households have three or more cars.

1) In a simple random sample of 400 car-owning households, determine the mean and standard

deviation number of car-owning households that will have three or more cars.

2) Interpret the mean.

µX = np and σ X = np 1− p( )

Of the 400 households, we would expect about 140 of them to have three or more cars.

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Objective – Check for unusual results in a Binomial Experiment

BIG IDEA: For a fixed probability of success, p, as the number of trials n in a binomial experiment increases, the

probability distribution of the random variable x becomes approximately

As a general rule of thumb, if , then the probability distribution in a binomial

experiment will be approximately bell-shaped. As a result, we can

Use the Empirical Rule to identify unusual observations in a binomial experiment.

Remember, the Empirical Rule states that in a bell-shaped distribution about 95% of all observations lie

within standard deviations of the mean.

Any observation that lies outside this interval may be considered because the

observation occurs less than of the time.

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Example: Using the Mean, Standard Deviation and Empirical Rule

to Check for Unusual Results in a Binomial Experiment

According to the Experian Automotive, 35% of all car-owning households have three or more cars. A

researcher believes the percentage is actually higher than that reported by Experian Automotive. He

conducts a simple random sample of 400 car-owning households and found that 162 had three or more

cars. Is this result unusual?

Analysis: We need to find out if the result, x = 162, is more than 2! away from the mean.

First, we need to see if the distribution is approximately bell-shaped by checking if

The result of 162 is unusual because it is more than two standard deviation's above the mean.

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