Stat 1220 - Ch 4.1 - 4.3 | 1

Section 4.1: Random Variables

Objectives:

Understand the concept of random variables

Distinguish between discrete random variables and continuous random variables

Random Variables

Random Variable - Represents a ___________________ generated by a random

experiment

Denoted by _____

(Note that we denote the actual values the random variables take by the lower

case letter x.)

Examples:

o X = Number of sales calls a salesperson makes in one day.

o X = Hours spent on sales calls in one day.

Discrete Random Variable - Has a finite or ____________ number of possible

outcomes that can be listed.

o Example: X = Number of sales calls a salesperson makes in one day.

Continuous Random Variable - Has an ______________ number of possible

outcomes, represented by an interval on the number line.

o Example: X = Hours spent on sales calls in one day.

Stat 1220 - Ch 4.1 - 4.3 | 2

*Important to distinguish between discrete and continuous random variables because

different statistical techniques are used to analyze each type.

Example: Discrete Variables and Continuous Variables

Decide whether the random variable x is discrete or continuous.

a) X = The number of Fortune 500 companies that lost money in the previous year.

Solution:

b) X = The volume of gasoline in a 21-gallon tank.

Solution:

c) X = The speed of the space shuttle.

Solution:

d) X = The number of calves born on a farm in one year.

Solution:

Have we met the objectives of the section???

Stat 1220 - Ch 4.1 - 4.3 | 3

Section 4.2: Probability Distributions for Discrete Random Variables

Objectives:

How to construct a discrete probability distribution and its graph

How to determine if a distribution is a probability distribution

How to find the mean, variance, and standard deviation of a discrete probability distribution

Discrete Probability Distributions

Discrete probability distribution -

o Lists each possible value the random variable can assume, together with its

probability.

o Must satisfy the following conditions:

In Words In Symbols 1. The probability of each value of the discrete random variable is between _________________, inclusive.

2. The sum of all the probabilities is ___.

Guidelines for Constructing a Discrete Probability Distribution:

Let X be a discrete random variable with possible outcomes x1, x2, … , xn.

1. Make a __________________________ for the possible outcomes.

2. Find the _______ of the frequencies.

3. Find the ________________ of each possible outcome by dividing

its frequency by sum of the frequencies.

4. Check that each probability is _____________________, inclusive,

and that the sum of all probabilities is_____.

Stat 1220 - Ch 4.1 - 4.3 | 4

Example: Constructing and Graphing a Discrete Probability Distribution

An industrial psychologist administered a personality inventory test for passive-aggressive traits

to 150 employees. Individuals were given a score from 1 to 5, where 1 was extremely passive

and 5 extremely aggressive. The results are shown below. A score of 3 indicated neither trait.

Construct a probability distribution for the random variable X.

Score, X Frequency, f

1 24

2 33

3 42

4 30

5 21

Solution:

Example: Verify the following distribution is a probability distribution.

Score, x Relative Frequency

0 0.216

1 0.432

2 0.288

3 0.064

Solution:

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Example: Identifying Probability Distributions

Decide whether the distribution is a probability distribution. Explain your reasoning.

1)

2)

Example: Recall Ch3 Info and Put Together with Property of Probability Distribution

Use the table below to answer the following questions.

X 2 5 7 9 11

P(x) .1 .2 .2 .3

a) Find the missing probability in the table.

b) Find P(X ≥ 7).

c) Find P(X ≤ 5 or X > 9)

X 5 6 7 8

P(x) 0.28 0.21 0.43 0.15

X 1 2 3 4

P(x) 1/2 1/4 5/4 -1

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Mean of a Discrete Random Variable

Mean of a discrete random variable X (also called the expected value):

o ______________________

(Each value of x is multiplied by its corresponding probability and the

products are added.)

The mean of a random variable represents the __________________________ of a

probability experiment and is sometimes not a possible outcome.

If the experiment was performed thousands of times, the mean of all the outcomes

would be close to the mean of the __________________________.

Example: Finding the Mean of a Probability Distribution

The probability distribution for the personality inventory test for passive-aggressive traits

is given. Find the mean score. (Recall 1 is extremely passive, 5 is extremely aggressive)

X P(x)

1 0.16

2 0.22

3 0.28

4 0.20

5 0.14

Solution:

Interpretation: The mean score is approximately _______. Recall from the first

example of this data set that a score of 3 represents an individual who exhibits

neither passive nor aggressive traits. You can conclude that the mean personality

trait is ______________________________________________________________

___________________________________________________________________.

The mean of a random variable of a probability distribution describes a typical

outcome, but it gives no info about how the outcomes ________.

Stat 1220 - Ch 4.1 - 4.3 | 7

Discrete Probability Distributions (Continued)

Variance of a discrete probability distribution:

∑( ) ( )

Standard deviation of a discrete probability distribution

√ √∑( ) ( )

Example: Finding the Variance and Standard Deviation

The probability distribution for the personality inventory test for passive-aggressive traits

is given. Find the variance and standard deviation. ( Recall μ = 2.94)

X P(x)

1 0.16

2 0.22

3 0.28

4 0.20

5 0.14

Solution:

Interpretation: Most of the data values differ from the mean by no more than ______.

Additional Comments about the Mean aka Expected Value:

The mean of a random variable represents what you would expect to happen over

thousands of trials.

Probabilities ______________ be negative, but the expected value of a random

variable _______ be negative.

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Example: Finding an Expected Value

At a raffle, 1500 tickets are sold at $2 each for four prizes of $500, $250, $150, and $75.

You buy one ticket. What is the expected value of your gain?

Solution:

Interpretation: You can expect to ______ an average of ______ for each ticket you buy.

Additional Example of Constructing a Probability Distribution

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

a) Construct the probability distribution of X

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

Have we met the objectives of the section???

Stat 1220 - Ch 4.1 - 4.3 | 9

Section 4.3: The Binomial Distribution

Objectives:

To learn the concept of a binomial random variable

Find the mean and standard deviation of a binomial random variable

Use the probability formula for a binomial random variable

Understand the cumulative probability distribution of a binomial random variable

Binomial random variable - The discrete random variable that counts the number

of _______________ in a fixed number of independent identical trials of a

success/failure experiment for which the probability of success on each trial is the

_______ number p.

"Successes" are the outcomes that are being counted; not necessarily which

outcome we think is "better."

o EX: A random sample of 125 students is selected from a large college in

which the proportion of students who are females is 57%. Suppose X

denotes the number of female students in the sample.

o EX: A test has 15 questions, each of which has five answer choices. An

unprepared student answers each of the questions randomly by choosing an

arbitrary answer from the five provided. Suppose X denotes the number of

answers that the student gets right.

o EX: An experimental medication was given to 30 patients with a certain

medical condition. Suppose X denotes the number of patients who develop

severe side effects.

Special Formulas for the Mean and Standard Deviation of a Binomial Random Variable

• There are special formulas for the mean, variance, and standard deviation of the

binomial random variable with parameters n and p that are much simpler than the

general formulas that apply to all discrete random variables.

• If X is a binomial random variable with parameters n and p, then

√

where q = 1 − p.

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Example: Seventeen percent of victims of financial fraud know the perpetrator of the

fraud personally. Let X be the number of people in a random sample of five victims of

financial fraud who knew the perpetrator personally. Find the mean, variance, and

standard deviation of the random variable X using the new short formulas for binomial

random variables.

Factorials: 0! =

1! =

2! =

3! =

4! =

n! =

Example:

a) 5! =

b)

=

c) ( )

=

Formula to calculate the probability p of a binomial random variable:

If X is a binomial random variable with parameters n and p, then

( )

( )

where q = 1 − p.

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Example: Seventeen percent of victims of financial fraud know the perpetrator of the

fraud personally

a) Use the formula to construct the probability distribution for the number X of

people in a random sample of five victims of financial fraud who knew the

perpetrator personally.

b) An investigator examines five cases of financial fraud every day. Find the most

frequent number of cases each day in which the victim knew the perpetrator.

c) An investigator examines five cases of financial fraud every day. Find the average

number of cases per day in which the victim knew the perpetrator.

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The Cumulative Probability Distribution of a Binomial Random Variable

• The probability entered in the table corresponds to the area of the shaded region

(see separate handout of cumulative binomial probability tables).

• The cumulative table is much easier to use for computing P(X ≤ x) since all the

individual probabilities have already been computed and added.

Example: A student takes a 10-question true/false exam.

a) Find the probability a student scores a 50% or lower just by guessing.

b) Find the probability that the student gets exactly six of the questions right simply

by guessing the answer on every question.

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c) Find the probability that the student will obtain a passing grade of 60% or greater

simply by guessing.

• Note: The one table suffices for both P(X ≤ x) or P(X ≥ x) and can be used to readily

obtain probabilities of the form P(x), too, because of the following formulas:

P(X ≥ x) = 1 − P (X ≤ x − 1)

P(X = x) = P(X ≤ x) − P(X ≤ x − 1)

Have we met the objectives of the section???

Stat 1220 - Ch 4.1 - 4.3 | 14 Additional Examples from Old Final Exams:

Suppose that 8% of all males suffer some form of color blindness. Find the probability that in a random

sample of 250 men at least 10% will suffer some form of color blindness.