Chapter 5 Probability Distributions (Discrete Variables)

Statistics I
MTH160
Chapter 5
Probability Distributions
(Discrete Variables)
5.1 Random Variables
5.2 Probability Distributions of a Discrete Random Variable
5.3 The Binomial Probability Distribution

MTH 160
Statistics I
Brigitte Martineau Chapter 5
What is a random variable?
5.1 Random Variables
Notes:
Examples of Random Variables
Let the number of computers sold per day by a local merchant be a random variable.
Integer values ranging from zero to about 50 are possible values.
Let the time it takes an employee to get to work be a random variable. Possible values
are 15 minutes to over 2 hours.
Let the volume of water used by a household during a month be a random variable.
Amounts range up to several thousand gallons.
Let the number of defective components in a shipment of 1000 be a random variable.
Values range from 0 to 1000.
Discrete versus Continuous Random Variables
Discrete Random Variables:
A quantitative random variable that can assume a __________________ number of values.
Also known as a ___________________
Continuous Random Variables:
A quantitative random variable that can assume an _______________________ number of
values.
Also known as a ____________________
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MTH 160
Statistics I
Brigitte Martineau Chapter 5
5.2 Probability Distributions of a Discrete Random Variable
Let’s start with an example.
Let’s say we are looking at the random variable x = Number of tails when 2 coins are tossed
What are the possible values for x?
Let’s make a table with all the possible values of x and their respective probability:
P(x = 0) P( zero tail) =
P(x = 1) P( one tail) =
P(x = 2) P (two tails) =
The table we just created is called a _______________________ ____________________
What is a Probability Distribution?
A ___________________ of the probabilities associated with each __________ of a
random variable.
The probability distribution is a ___________________ distribution
It is used to represent ____________________
Note:
What is a Probability Function?
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MTH 160
Statistics I
Brigitte Martineau Chapter 5
Examples:
The probability distribution of a modified die is as follow: P( x) x / 6 for x 1,2,3
X 1 2 3
P(x)
Could you find a probability distribution that describes the probabilities obtained when
rolling a regular die?
X 1 2 3 4 5 6
P(x)
Properties of Probability Distribution
Property 1: 0 Px
( ) 1
Property 2: Px ( ) 1
all x
Examples:
The number of people staying in a randomly selected room at a local hotel is a discrete
random variable ranging in value from 0 to 4. The probability distribution is known and
given in the form of a chart below.
X 0 1 2 3 4
P(x) 1/15 2/15 3/15 4/15 5/15
a) Verify that this distribution meets the 2 properties of a probability distribution.
b) Express the distribution as a probability function.
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MTH 160
Statistics I
Brigitte Martineau Chapter 5
c) Construct a histogram of the Hotel Room probability distribution.
NOTES:
‣ The histogram of a probability distribution uses the physical ________ of each
bar to represent its assigned probability
‣ In the Hotel Room probability distribution: the width of each bar is ______, so
the height of each bar is equal to the assigned probability, which is the area of
each bar.
‣ The idea of area representing probability is important in the study of
___________________ _______________ variables
x
Is P( x) for x 1,2,3,4 a probability function? Why?
9
x
P(x)
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MTH 160
Statistics I
Brigitte Martineau Chapter 5
Mean and Variance of Discrete Probability Distribution
Goal:
To describe the ________________ and the _______________________ of a population
Let’s review first!
1. The mean of a sample is represented by ___________.
2. s 2 and s are the ____________ and _____________ ____________ of the _________.
3. s 2 , and s are called ____________ _________________.
4. µ (lowercase Greek letter “mu”) is the _______________ of the _________________
5.
2
(“sigma squared”) is the ______________ of the________________.
6. σ (lowercase Greek letter “sigma”) is the _____________ _________________ of the
_________________ ___.
2
7. µ, σ and are called _______________ ________________.
2
(A parameter is a constant. µ, σ and are typically unknown values.)
The Mean of a Discrete Random Variable
The mean, µ, of a discrete random variable x is found by ____________________ each
possible value of x by its own probability and then adding all the _________________ together.
xP( x)
Notes:
The mean is not necessarily a value of the random variable
The mean ________ is a population parameter that is usually unknown. We would like to
____________________ its value.
The Variance of a Discrete Random Variable
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MTH 160
Statistics I
Brigitte Martineau Chapter 5
2
The variance, , of a discrete random variable x is found by _________________ each
possible value of the squared deviation from the mean,
2
( x )
, by its own probability and then
_____________ all the products together.
2
x
2
x P x
2
P( x)
( )
xP( x)
2 2
x P( x)
2
The Standard Deviation of a Discrete Random Variable
2
Examples:
Find the mean, the variance and the standard deviation of the following probability
x
distribution: P( x) for x 1 and 2
3
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MTH 160
Statistics I
Brigitte Martineau Chapter 5
The number of standby passengers who get seats on a daily commuter flight from
Boston to New York is a random variable, x, with probability distribution given below. Find
the mean, variance, and standard deviation.
x
P(x)
0 0.30
1 0.25
2 0.20
3 0.15
4 0.05
5 0.05
Total
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MTH 160
Statistics I
Brigitte Martineau Chapter 5
5.3 - The Binomial Probability Distribution
Before we start this section, let us review the concept of factorial.
Factorial means the product of a particular set of integers and denoted by !
4!
12!
x!
0!
Suppose you are given a three-question multiple-choice quiz. You have missed the last week of
class and have not read the material in your textbook. You decide to take the quiz and just
randomly guess all the answers. Let’s take the quiz:
1. A B C D
2. A B C D
3. A B C D
Are questions 1, 2 and 3 independent or dependent events? ______________________
How many questions do you think you answered correctly? _______________________
What do you think is the probability that you got 100% on the quiz? _________________
What do you think is the probability that you got 0% on the quiz? ___________________
What do you think would be the class average? ________________________________
Let’s look at all the possibilities using a tree diagram:
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MTH 160
Statistics I
Brigitte Martineau Chapter 5
Question:
For each individual question, what is the probability that you got the right answer? ___
The probability that you got the wrong answer? ___ Why?
Let x = a random variable representing the number of correct answers.
P(x = 0) =
P(x = 1) =
P(x = 2) =
P(x = 3) =
This last experiment is known as a Binomial Probability Experiment
What is a Binomial Probability Experiment?
An experiment that is made of _________________ trials that possess the following properties:
There are _____ repeated independent trials
Each trial has two possible outcomes: a _________________ or a ______________
P(success) = _______, P(failure) = __________ and ___ ___ 1
The binomial random variable x is the _________ of the number of successful trials that
occur; x may take any integer value from 0 to n.
Examples
1) Let’s look at our quiz
Trials
n :
Success: Failure:
p : q :
x :
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MTH 160
Statistics I
Brigitte Martineau Chapter 5
2) A die is rolled 20 times and the number of “fives” that occurred is reported as the random
variable. Explain why x is a binomial random variable.
Trials
n :
Success: Failure:
p : q :
x :
The Binomial Probability Function
n
x nx
P( x) p q , for x 0,1,2,3,...
x
What is n
?
x
What is
p x ?
What is q n x ?
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MTH 160
Statistics I
Brigitte Martineau Chapter 5
Examples:
Back to our quiz, let us use the formula to find:
Px ( 0)
Px ( 2)
Px ( 1)
Px ( 3)
Results from the 2000 census show that 42% of U.S. grandparents are the primary
caregivers for their grandchildren (Democrat & Chronicle, “Grandparents as ma and pa,”
July 8, 2002). In a group of 20 grandparents, what is the probability that exactly half are
primary caregivers for their grandchildren? At most 19 are primary caregivers for their
grandchildren?
First let us check if this is a Binomial experiment.
Trials n :
Success:
Failure:
p : q :
x :
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MTH 160
Statistics I
Brigitte Martineau Chapter 5
Using Tables (p. 807 - 809)
It is also possible to use a table to compute the probabilities of a Binomial Experiment as long
as n 15 and that p is listed in the table (0.01, 0.05, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9,
0.95 and 0.99)
Let’s use the table to answer the following questions:
The survival rate during a risky operation for patients with no other hope of survival is
80%. What is the probability that exactly four of the next five patients survive this
operation?
Success: p : n:
Failure: q :
If boys and girls are equally likely to be born, what is the probability that in a randomly
selected family of six children, that there will be:
Success: p : n:
Failure: q :
Exactly 4 boys
Exactly 2 girls
At least 3 boys
At most 2 boys
At most 5 boys
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MTH 160
Statistics I
Brigitte Martineau Chapter 5
Mean and Standard Deviation of the Binomial Distribution
The mean and the standard deviation of the binomial distribution are as follow:
Examples:
Mean
np
S tan dard Deviation
npq
2
Variance
Find the mean and the standard deviation for the number of sixes seen in 50 rolls of a die.
The probability of success on a single trial of a binomial experiment is known to be ¼. The
random variable x , number of successes, has a mean value of 80. Find the number of trials
involved in this experiment and the standard deviation of x .
Consider the binomial distribution where n 4 and p 0.3
o Find the mean
o Find the standard deviation
o Using Table 2, find the probability distribution and draw a histogram.
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