Transcript Business Statistics: A Decision

Business Statistics:
A Decision-Making Approach
8th Edition
Chapter 5
Discrete Probability Distributions
5-1
Chapter Goals
After completing this chapter, you should be able to:

Calculate and interpret the expected value of a discrete
probability distribution

Apply the binomial distribution to business problems

Compute probabilities for the Poisson and hypergeometric
distributions

Recognize when to apply discrete probability distributions
to decision making situations
5-2
Probability Distributions
Ch. 5
Ch. 6
Discrete
Probability
Distributions
Continuous
Probability
Distributions
Binomial
Normal
Poisson
Uniform
Hypergeometric
Exponential
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Random variable vs. Probability distribution


When the value of a variable is the outcome of
a statistical experiment, that variable is a random
variable.
A probability distribution is a table or an equation that
links each outcome of a statistical experiment with its
probability of occurrence.
5-4
Random variable vs. Probability distribution
Experiment: Toss 2 Coins.
Let x = # heads.
4 possible outcomes
T
T
T
H
H
T
H
H
x Value
Probability
0
1/4 = 0.25
1
2/4 = 0.50
2
1/4 = 0.25
5-5
Cumulative Probability and
Cumulative Probability Distribution

A cumulative probability refers to the
probability that the value of a random variable
falls within a specified range.

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Probability for at most (less than equal to: <) one
head?  0.25+0.5=0.75
A cumulative probability distribution can be
represented by a table or an equation.
5-6
Cumulative Probability Distribution
Cumulative
Number of heads: x Probability: P(X = x) Probability:
P(X < x)
0
0.25
0.25
1
0.50
0.75
2
0.25
1.00
5-7
Discrete Probability Distribution

If a random variable is a discrete variable,
its probability distribution is called a discrete
probability distribution.
Number of heads
Probability
0
0.25
1
0.50
2
0.25
5-8
Mean (formula)

Expected Value (or mean) of a discrete probability distribution
(Weighted Average – e.g., GPA)
E(x) = xP(x)

Example: Toss 2 coins,
x = # of heads,
compute expected value of x:
x
P(x)
0
0.25
1
0.50
2
0.25
E(x) = (0 x 0.25) + (1 x 0.50) + (2 x 0.25)
= 1.0
5-9
Standard Deviation (formula)

Standard Deviation of a discrete probability distribution
σx 
{x  E(x)}
2
P(x)
where:
E(x) = Expected value of the random variable (done!)
x = Values of the random variable
P(x) = Probability of the random variable having
the value of x
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Standard Deviation
(continued)

Example: Toss 2 coins, x = # heads,
compute standard deviation (recall E(x) = 1)
σx 
 {x  E(x)}
2
P(x)
σ x  (0  1) 2 (0.25)  (1  1) 2 (0.50)  (2  1) 2 (0.25)  0.50  .707
Possible number of heads
= 0, 1, or 2
5-11
Using Excel


Review real world business examples on page 194 and
page 195
Use Excel for calculating:

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Discrete Random Variable Mean
Discrete Random Variable Standard Deviation
Download and open “Binomial Poisson Distribution”
Excel file…

And then, try the example on the first tap…
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Binomial Experiment



The experiment involves repeated trials.
Each trial has only two possible outcomes - a
success or a failure (i.e., head/tail, goal/no goal).
The probability that a particular outcome will occur on
any given trial is constant.

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0.5 every trial
All of the trials in the experiment are independent.

The outcome on one trial does not affect the outcome on other
trials.
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Binomial Experiment Example
Outcome, x
Binomial
probability,
P(X = x)
Cumulative
probability,
P(X < x)
0 Heads
0.125
0.125
1 Head
0.375
0.500
2 Heads
0.375
0.875
3 Heads
0.125
1.000
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Binomial Probability

A binomial probability refers to the probability of getting
EXACTLY n successes in a specific number of trials.

Example: What is the probability of getting EXACTLY 2
Heads in 3 coin tosses.

Using the table on the previous slide, that probability (0.375)
would be an example of a binomial probability.
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Cumulative Binomial Probability


Cumulative binomial probability refers to the probability
that the value of a binomial random variable falls within
a specified range.
Example: What is the probability of getting AT MOST 2
Heads (meaning, less than equal to: <) in 3 coin tosses
is an example of a cumulative probability.

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0 heads (0.125) + 1 head (0.375) + 2 heads (0.375).
Thus, the cumulative probability of getting AT MOST 2 Heads in
3 coin tosses is equal to 0.875.
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Translation to Math Notations


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The probability of getting FEWER (LESS) THAN 2
successes is indicated by P(X < 2).
The probability of getting AT MOST 2 successes is
indicated by P(X < 2).
The probability of getting AT LEAST 2 successes is
indicated by P(X > 2).
The probability of getting MORE (GREATER) THAN 2
successes is indicated by P(X > 2).
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Binomial Distribution

The shape of the binomial distribution depends on the
values of p and n
Mean

Here, n = 5 and p = 0.1
Try the “Binomial
Distribution Simulation”
on the class website

Here, n = 5 and p = 0.5
P(X)
.6
.4
.2
0
n = 5 p = 0.1
X
0
P(X)
.6
.4
.2
0
1
2
3
4
5
n = 5 p = 0.5
X
0
1
2
3
4
5
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Binomial Distribution Formula
n!
x nx
P(x) 
p q
x ! (n  x ) !
P(x) = probability of x successes in n trials,
with probability of success p on each trial
x = number of successes in sample,
(x = 0, 1, 2, ..., n)
p = probability of “success” per trial
q = probability of “failure” = (1 – p)
n = number of trials (sample size)
5-19
Binomial Distribution Example

Example: 35% of all voters support Proposition
A. If a random sample of 10 voters is polled,
what is the probability that exactly three of them
support the proposition?
i.e., find P(x = 3) if n = 10 and p = 0.35 :
n!
10!
x n x
P(x  3) 
p q 
(0.35) 3 (0.65) 7  0.2522
x! (n  x)!
3!7!
There is a 25.22% chance that 3 out of the 10 voters
will support Proposition A
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Using Excel

Try the binominal distribution using Excel…

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Download and open “Binomial Poisson Distribution”
Excel file…
Try followings together;
 Binom-1
 Binom-2
 Binom-3
Then, try exercise 5-34 and 5-36 with your neighboor
5-21
The Poisson Distribution


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The Poisson Distribution is a discrete distribution
which takes on the values X = 0, 1, 2, 3, ... .
It is often used as a model for the number of
events in a specific time period.
Events examples:


the number of telephone calls at a call center
the number of bags lost per flight
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Poisson Distribution
Summary Measures


Mean
μ  λt
Variance and Standard Deviation
σ  λt
2
σ  λt
where
 = number of successes in a segment of unit size
t = the size of the segment of interest
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Poisson Distribution Formula
( t ) e
P( x ) 
x!
x
 t
where:
t = size of the segment of interest
x = number of successes in segment of interest
 = expected number of successes in a segment of unit size
e = base of the natural logarithm system (2.71828...)
Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall
5-24
Example


The average number of homes sold by the Acme Realty
company is 2 homes per day. What is the probability
that exactly 3 homes will be sold tomorrow?
Solution: This is a Poisson experiment in which we
know the following:


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μ = 2; since 2 homes are sold per day, on average.
x = 3; since we want to find the likelihood that 3 homes will be
sold tomorrow.
e = 2.71828; since e is a constant equal to approximately
2.71828.
5-25
Example (con’t)

We plug these values into the Poisson formula as
follows:
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P(3; 2) = (23) (2.71828-2) / 3!
P(3; 2) = (0.13534) (8) / 6
P(3; 2) = 0.180
Thus, the probability of selling 3 homes tomorrow is
0.180 .
5-26
Poisson distribution – Using Excel
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Excel can be used to find both the cumulative
probability as well as the point estimated probability for
a Poisson experiment.
In order to get Excel to calculate poisson probabilities,
you have to use the following syntax in a cell.
=poisson (x; mean; cumulative)
Previous example

=poisson (2; 3; false) = 0.180
5-27
Poisson distribution – Using Excel
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X is the number of events.
Mean is simply the mean of the variable.
Cumulative has the options of FALSE and TRUE.
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If you choose FALSE, Excel will return probability of only and
only the x number of events happening.
If you choose TRUE, Excel will return the cumulative probability
of the event x or less happening.
5-28
Example: Point Estimate
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A bakery has average 6 customers during a business
hour. The bakery wishes to calculate the probability of
the event that exactly 4 customers enter the store in the
next hour.
That is: x = 4, mean = 6 and cumulative = FALSE
Would be written in excel as: =poisson(4;6;FALSE)
And return the probability of 0.133853 = 13.3853%
5-29
Example: Cumulative
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
A bakery has average 6 customers during a business
hour. We then wish to calculate the probability of the
event that 4 customers or less enter the store in the
next hour.
That is: x = 4, mean = 6 and cumulative = TRUE
Would be written in excel as: =poisson(4;6;TRUE)
And return the probability of 0.285057 = 28.5057%
5-30
Using Excel

Try the poisson distribution using Excel…


Download and open “Binomial Poisson Distribution”
Excel file…
Try “Poisson”………
5-31
Poisson Distribution Heritage Title
Try this…….
Issue: The distribution for the number of
defects per tile made by Heritage Tile is Poisson
distributed with a mean of 3 defects per tile. The
manager is worried about the high variability
Objective: Use Excel 2007 or 2010 to generate
the Poisson distribution and histogram to
visually see spread in the distribution of
possible defects.
5-32
Poisson Distribution – Heritage Tile
Enter values zero
through 10
5-33
Poisson Distribution – Heritage Tile
Select Formulas,
More Functions,
Statistical and
POISSON
5-34
Poisson Distribution – Heritage Tile
Enter:
a1, 3, false
5-35
Poisson Distribution – Heritage Tile
Notice that I had preselected Cell B1.
When I pressed enter the
Poisson Probability was
loaded into that cell.
Simply copy and paste Cell
B1 into cells B2 : B11
5-36
Poisson Distribution – Heritage Tile
•Select the Insert tab
•Select Column
•Select the chart type
that you want
5-37
Poisson Distribution – Heritage Tile
Format the chart
as per Chapter 2
5-38
Chapter Summary

Reviewed key discrete distributions



Binomial
Poisson
Hypergeometric

Found probabilities using formulas and tables

Recognized when to apply different distributions

Applied distributions to decision problems
5-39