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SLIDES BY
John Loucks
St. Edward’s Univ.
© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 1
Chapter 3, Part A
Discrete Probability Distributions
Introduction to probability
Random Variables
Discrete Probability Distributions
Binomial Probability Distribution
Poisson Probability Distribution
.40
.30
.20
.10
0
1
© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
2
3
4
Slide 2
Uncertainties
Managers often base their decisions on an
analysis of uncertainties such as the following:
What are the chances that sales will decrease
if we increase prices?
What is the likelihood a new assembly method
will increase productivity?
What are the odds that a new investment will
be profitable?
© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 3
Probability
Probability is a numerical measure of the likelihood
that an event will occur.
Probability values are always assigned on a scale
from 0 to 1.
A probability near zero indicates an event is quite
unlikely to occur.
A probability near one indicates an event is almost
certain to occur.
© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 4
Probability as a Numerical Measure
of the Likelihood of Occurrence
Increasing Likelihood of Occurrence
Probability:
0
The event
is very
unlikely
to occur.
.5
The occurrence
of the event is
just as likely as
it is unlikely.
1
The event
is almost
certain
to occur.
© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 5
Statistical Experiments
In statistics, the notion of an experiment differs
somewhat from that of an experiment in the
physical sciences.
In statistical experiments, probability determines
outcomes.
Even though the experiment is repeated in exactly
the same way, an entirely different outcome may
occur.
For this reason, statistical experiments are sometimes called random experiments.
© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 6
Assigning Probabilities
◼ Basic Requirements for Assigning Probabilities
1. The probability assigned to each experimental
outcome must be between 0 and 1, inclusively.
0 < P(Ei) < 1 for all i
Where: Ei is the ith experimental outcome and P(Ei) is its
probability
2. The sum of the probabilities for all experimental
outcomes must equal 1.
P(E1) + P(E2) + . . . + P(En) = 1
Where: n is the number of experimental outcomes
© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 7
1. Random Variables
A random variable is a numerical description of the
outcome of an experiment.
A discrete random variable may assume either a
finite number of values or an infinite sequence of
values.
A continuous random variable may assume any
numerical value in an interval or collection of
intervals.
© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 8
Random Variables
Examples of Random Variables
The first, second, and fourth variables above are discrete,
while the third one is continuous.
© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 9
Random Variables
Examples of Random Variables
Question
Family
size
Type
Random Variable x
x = Number of dependents in
family reported on tax return
Discrete
Distance from x = Distance in miles from
home to store home to the store site
Continuous
Own dog
or cat
Discrete
x = 1 if own no pet;
= 2 if own dog(s) only;
= 3 if own cat(s) only;
= 4 if own dog(s) and cat(s)
© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 10
Example: JSL Appliances
Discrete random variable with a finite number of
values
Let x = number of TVs sold at the store in one day,
where x can take on 5 values (0, 1, 2, 3, 4)
© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 11
Example: JSL Appliances
Discrete random variable with an infinite sequence
of values
Let x = number of customers arriving in one day,
where x can take on the values 0, 1, 2, . . .
We can count the customers arriving, but there is no
finite upper limit on the number that might arrive.
© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 12
2. Discrete Probability Distributions
The probability distribution for a random variable
describes how probabilities are distributed over
the values of the random variable.
We can describe a discrete probability distribution
with a table, graph, or equation.
© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 13
Discrete Probability Distributions
The probability distribution is defined by a
probability function, denoted by f(x), which provides
the probability for each value of the random variable.
The required conditions for a discrete probability
function are:
f(x) > 0
f(x) = 1
For a discrete probability distribution we calculate
the probability of being less than some value x, i.e.
P(X < x), by simply summing up the probabilities of
the values less than x.
© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 14
Example: DiCarlo Motors, Inc.
Using past data on daily car sales, …
a tabular representation of the probability
distribution for car sales was developed.
x
0
1
2
3
4
5
f(x)
.18
.39
.24
.14
.04
.01
1.00
© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 15
Example: DiCarlo Motors, Inc.
Graphical Representation of the Probability
Distribution
Probability
.50
.40
.30
.20
.10
0
1
2
3
4
5
Values of Random Variable x (car sales)
© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 16
Example: DiCarlo Motors, Inc.
The probability distribution provides the following
information.
• There is a 0.18 probability that no cars will be
sold during a day.
• The most probable sales volume is 1, with f(1) =
0.39.
• There is a 0.05 probability of an outstanding sales
day with four or five cars being sold.
© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 17
Discrete Uniform Probability Distribution
The discrete uniform probability distribution is the
simplest example of a discrete probability
distribution given by a formula.
The discrete uniform probability function is
f(x) = 1/n
the values of the
random variable
are equally likely
where:
n = the number of values the random
variable may assume
© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 18
Expected Value and Variance
The expected value, or mean, of a random variable
is a measure of its central location.
E(x) = = xf(x)
The variance summarizes the variability in the
values of a random variable.
Var(x) = 2 = (x - )2f(x)
The standard deviation, , is defined as the positive
square root of the variance.
© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 19
Example: DiCarlo Motors, Inc.
Expected Value of a Discrete Random Variable
x
0
1
2
3
4
5
f(x)
xf(x)
.18
.00
.39
.39
.24
.48
.14
.42
.04
.16
.01
.05
E(x) = 1.50
expected number of
cars sold in a day
© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 20
Example: DiCarlo Motors, Inc.
Variance and Standard Deviation
of a Discrete Random Variable
x
x-
(x - )2
0
1
2
3
4
5
.18
-1.5
2.25
.4050
.39
-0.5
0.25
.0975
.24
0.5
0.25
.0600
.14
1.5
2.25
.3150
.04
2.5
6.25
.2500
.01
3.5
12.25
.1225
Variance of daily sales = 2 = 1.2500
f(x)
(x - )2f(x)
cars
squared
Standard deviation of daily sales = 1.118 cars
© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 21
3. Binomial Probability Distribution
Four Properties of a Binomial Experiment
1. The experiment consists of a sequence of n
identical trials.
2. Two outcomes, success and failure, are possible
on each trial.
3. The probability of a success, denoted by p, does
not change from trial to trial.
stationarity
assumption
4. The trials are independent.
© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 22
Binomial Probability Distribution
Our interest is in the number of successes
occurring in the n trials.
We let x denote the number of successes
occurring in the n trials.
© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 23
Binomial Probability Distribution
Binomial Probability Function
n!
f (x) =
p x (1 − p)( n − x )
x !(n − x )!
where:
f(x) = the probability of x successes in n trials
p = the probability of success on any one trial
n = the number of trials
x = number of successes in n trials
n! =n( n - 1)( n - 2) . . . (2)(1)
© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 24
Binomial Probability Distribution
Binomial Probability Function
n!
f (x) =
p x (1 − p)( n − x )
x !(n − x )!
Number of experimental
outcomes providing exactly
x successes in n trials
Probability of a particular
sequence of trial outcomes
with x successes in n trials
© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 25
Example: Nastke Clothing Store
Binomial Probability Distribution
The store manager estimates that the probability
of a customer making a purchase is 0.30. What is the
probability that 2 of the next 3 customers entering the
store make a purchase?
Let: p = .30 (success), n = 3, x = 2
n!
f ( x) =
p x (1 − p ) (n − x )
x !( n − x )!
3!
f (2) =
(0.3)2 (0.7)1 = 3(.09)(.7) = .189
2!(3 − 2)!
© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 26
Binomial Probability Distribution
TABLE 3.6 PROBABILITY DISTRIBUTION FOR THE
NUMBER OF CUSTOMERS
MAKING A PURCHASE
X
f (x)
0
0.343
1
0.441
2
0.189
3
0.027
Total 1.000
© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 27
Binomial Probability Distribution
Expected Value
E(x) = = np
Variance
Var(x) = 2 = np(1 − p)
Standard Deviation
s = np(1- p)
© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 28
Example: Nastke Clothing Store
Binomial Probability Distribution
• Expected Value
E(x) = = 3(.3) = .9 customers out of 3
• Variance
Var(x) = 2 = 3(.3)(.7) = .63
• Standard Deviation
= 3(.3)(.7) = .794 customers
© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 29
Poisson
Probability
Distribution
Poisson
Probability Distribution
A Poisson distributed random variable is often
useful in estimating the number of occurrences
over a specified interval of time or space
It is a discrete random variable that may assume
an infinite sequence of values (x = 0, 1, 2, . . . ).
© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 30
Poisson
Probability
Distribution
Poisson
Probability Distribution
Examples of Poisson distributed random variables:
the number of knotholes in 14 linear feet of
pine board
the number of vehicles arriving at a toll
booth in one hour
Bell Labs used the Poisson distribution to model
the arrival of phone calls.
© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 31
Poisson
Probability
Distribution
Poisson
Probability Distribution
Two Properties of a Poisson Experiment
1. The probability of an occurrence is the same
for any two intervals of equal length.
2. The occurrence or nonoccurrence in any
interval is independent of the occurrence or
nonoccurrence in any other interval.
© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 32
Poisson Probability Distribution
Poisson Probability Function
𝜇 𝑥 𝑒 −𝜇
𝑓 𝑥 =
𝑥!
where:
x = the number of occurrences in an interval
f(x) = the probability of x occurrences in an interval
= mean number of occurrences in an interval
e = 2.71828
x! = x(x – 1)(x – 2) . . . (2)(1)
© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 33
Poisson Probability Distribution
Poisson Probability Function
Since there is no stated upper limit for the number
of occurrences, the probability function f(x) is
applicable for values x = 0, 1, 2, … without limit.
In practical applications, x will eventually become
large enough so that f(x) is approximately zero
and the probability of any larger values of x
becomes negligible.
© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 34
Poisson Probability Distribution
Example: Mercy Hospital
Patients arrive at the emergency room of Mercy
Hospital at the average rate of 6 per hour on
weekend evenings.
What is the probability of 4 arrivals in 30 minutes
on a weekend evening?
© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 35
Poisson
Probability
Distribution
Poisson
Probability Distribution
Example: Mercy Hospital
= 6/hour = 3/half-hour, x = 4
𝑓 4 =
34 (2.71828)−3
4!
Using the
probability
function
= .1680
© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 36
Chapter 3, Part B
Continuous Probability Distributions
Uniform Probability Distribution
Normal Probability Distribution
Exponential Probability Distribution
f (x)
f (x) Exponential
Uniform
f (x)
Normal
x
x
x
© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 37
Continuous Random Variables
Examples of continuous random variables include the
following:
• The flight time of an airplane traveling from
Chicago to New York
• The lifetime of the picture tube in a new television
set
• The drilling depth required to reach oil in an
offshore drilling operation
© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 38
Continuous Probability Distributions
A continuous random variable can assume any value
in an interval on the real line or in a collection of
intervals.
It is not possible to talk about the probability of the
random variable assuming a particular value.
Instead, we talk about the probability of the random
variable assuming a value within a given interval.
For a continuous probability distribution we calculate
the probability of being less than some value x, i.e.
P(X < x), by calculating the area under the curve to
the left of x.
© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 39
Continuous Probability Distributions
The probability of the random variable assuming a
value within some given interval from x1 to x2 is
defined to be the area under the graph of the
probability density function between x1 and x2.
f (x)
f (x) Exponential
Uniform
f (x)
x1 x 2
Normal
x
x1
x1 x2
xx12 x2
x
x
© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 40
Normal Probability Distribution
The normal probability distribution is the most
important distribution for describing a continuous
random variable.
It is widely used in statistical inference.
© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 41
Normal Probability Distribution
It has been used in a wide variety of applications:
Heights
of people
Test
scores
Amounts
of rainfall
Scientific
measurements
© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 42
Normal Probability Distribution
Normal Probability Density Function
f (x) =
1
s 2p
e
-(x-m )2 /2s 2
where:
= mean
= standard deviation
= 3.14159
e = 2.71828
© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 43
Normal Probability Distribution
Characteristics
The distribution is symmetric, and is bell-shaped.
x
© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 44
Normal Probability Distribution
Characteristics
The entire family of normal probability
distributions is defined by its mean and its
standard deviation .
Standard Deviation
Mean
x
© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 45
Normal Probability Distribution
Characteristics
The highest point on the normal curve is at the
mean, which is also the median and mode.
x
© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 46
Normal Probability Distribution
Characteristics
The mean can be any numerical value: negative,
zero, or positive.
x
-10
0
20
© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 47
Normal Probability Distribution
Characteristics
The standard deviation determines the width of the
curve: larger values result in wider, flatter curves.
= 15
= 25
x
© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 48
Normal Probability Distribution
Characteristics
Probabilities for the normal random variable are
given by areas under the curve. The total area
under the curve is 1 (.5 to the left of the mean and
.5 to the right).
.5
.5
x
© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 49
Normal Probability Distribution
Characteristics
68.26% of values of a normal random variable
are within +/- 1 standard deviation of its mean.
95.44% of values of a normal random variable
are within +/- 2 standard deviations of its mean.
99.72% of values of a normal random variable
are within +/- 3 standard deviations of its mean.
© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 50
Normal Probability Distribution
Characteristics
99.72%
95.44%
68.26%
– 3
– 1
– 2
+ 3
+ 1
+ 2
© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
x
Slide 51
Standard Normal Probability Distribution
-Use probability tables that have been calculated on a
computer.
- Only one special Normal distribution, N(0, 1), has
been tabulated.
- If we want to calculate probabilities from different
Normal distributions we convert the probability to one
involving the standard Normal distribution.
→ standardization
A random variable having a normal distribution
with a mean of 0 and a standard deviation of 1 is
said to have a standard normal probability
distribution.
© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 52
Standard Normal Probability Distribution
The letter z is used to designate the standard
normal random variable.
=1
z
0
© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 53
Standard Normal Probability Distribution
Converting to the Standard Normal Distribution
z=
x−
We can think of z as a measure of the number of
standard deviations x is from .
© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 54
Example: Pep Zone
Standard Normal Probability Distribution
Pep Zone sells auto parts and supplies including
a popular multi-grade motor oil. When the stock of
this oil drops to 20 gallons, a replenishment order is
placed.
© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 55
Example: Pep Zone
Standard Normal Probability Distribution
The store manager is concerned that sales are
being lost due to stockouts while waiting for an
order. It has been determined that demand during
replenishment lead-time is normally distributed with
a mean of 15 gallons and a standard deviation of 6
gallons.
The manager would like to know the probability
of a stockout, P(x > 20).
© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 56
Example: Pep Zone
Solving for the Stockout Probability
Step 1: Convert x to the standard normal distribution.
z = (x - )/
= (20 - 15)/6
= .83
Step 2: Find the area under the standard normal
curve between the mean and z = .83.
see next slide
© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 57
Example: Pep Zone
Probability Table for the
Standard Normal Distribution
z
.00
.01
.02
.03
.04
.05
.06
.07
.08
.09
.
.
.
.
.
.
.
.
.
.
.
.5
.1915 .1695 .1985 .2019 .2054 .2088 .2123 .2157 .2190 .2224
.6
.2257 .2291 .2324 .2357 .2389 .2422 .2454 .2486 .2517 .2549
.7
.2580 .2611 .2642 .2673 .2704 .2734 .2764 .2794 .2823 .2852
.8
.2881 .2910 .2939 .2967 .2995 .3023 .3051 .3078 .3106 .3133
.9
.3159 .3186 .3212 .3238 .3264 .3289 .3315 .3340 .3365 .3389
.
.
.
.
.
.
.
.
.
.
.
P(0 < z < .83)
© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 58
Example: Pep Zone
Solving for the Stockout Probability
Step 3: Compute the area under the standard normal
curve to the right of z = .83.
P(z > .83) = .5 – P(0 < z < .83)
= .5- .2967
= .2033
Probability
of a stockout
P(x > 20)
© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 59
Example: Pep Zone
Solving for the Stockout Probability
Area = .5 - .2967
Area = .2967
= .2033
0
.83
z
© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 60
End of Chapter 3
© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 61