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Business Statistics: A Decision-Making Approach

8th Edition

Chapter 5Discrete Probability Distributions

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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

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Probability Distributions

Continuous Probability

Distributions

Binomial

Hypergeometric

Poisson

Discrete Probability

Distributions

Normal

Uniform

Exponential

Ch. 5 Ch. 6

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.

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Experiment: Toss 2 Coins. Let x = # heads.

T

T

Random variable vs. Probability distribution

4 possible outcomes

T

T

H

H

H H

x Value Probability

0 1/4 = 0.25

1 2/4 = 0.50

2 1/4 = 0.25

Cumulative Probability andCumulative Probability Distribution

A cumulative probability refers to the probability that the value of a random variable falls within a specified range. 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.

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Cumulative Probability Distribution

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Number of heads: x Probability: P(X = x)Cumulative Probability: P(X < x)

0 0.25 0.25

1 0.50 0.75

2 0.25 1.00

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If a random variable is a discrete variable, its probability distribution is called a discrete probability distribution.

Discrete Probability Distribution

Number of heads Probability

0 0.25

1 0.50

2 0.25

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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:

E(x) = (0 x 0.25) + (1 x 0.50) + (2 x 0.25) = 1.0

x P(x)

0 0.25

1 0.50

2 0.25

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Standard Deviation of a discrete probability distribution

where:

E(x) = Expected value of the random variable (done!) x = Values of the random variableP(x) = Probability of the random variable having

the value of x

Standard Deviation (formula)

P(x)E(x)}{xσ 2x

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Example: Toss 2 coins, x = # heads, compute standard deviation (recall E(x) = 1)

Standard Deviation

P(x)E(x)}{xσ 2x

.7070.50(0.25)1)(2(0.50)1)(1(0.25)1)(0σ 222x

(continued)

Possible number of heads = 0, 1, or 2

Using Excel

Review real world business examples on page 194 and page 195

Use Excel for calculating: 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. 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

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Outcome, xBinomial 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

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.

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 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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n = 5 p = 0.1

n = 5 p = 0.5

Mean

0.2.4.6

0 1 2 3 4 5

X

P(X)

.2

.4

.6

0 1 2 3 4 5

X

P(X)

0

Binomial Distribution

The shape of the binomial distribution depends on the values of p and n

Here, n = 5 and p = 0.1

Here, n = 5 and p = 0.5

Try the “Binomial Distribution Simulation”

on the class website

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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)

P(x)n

x ! n xp qx n x!

( ) !

Binomial Distribution Formula

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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 :

.25220(0.65)(0.35)3!7!

10!qp

x)!(nx!

n!3)P(x 73xnx

There is a 25.22% chance that 3 out of the 10 voters will support Proposition A

Using Excel

Try the binominal distribution using Excel… 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

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The Poisson Distribution

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

Variance and Standard Deviation

λtμ

λtσ2

λtσ where = number of successes in a segment of unit size

t = the size of the segment of interest

Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 5-24

Poisson Distribution Formula

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...)

!x

e)t()x(P

tx

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:

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.

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Example (con’t)

We plug these values into the Poisson formula as follows:

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 .

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Poisson distribution – Using Excel

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

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Poisson distribution – Using Excel

X is the number of events. Mean is simply the mean of the variable. Cumulative has the options of FALSE and TRUE.

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.

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Example: Point Estimate

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 = FALSEWould be written in excel as: =poisson(4;6;FALSE)

And return the probability of 0.133853 = 13.3853%

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Example: Cumulative

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 = TRUEWould be written in excel as: =poisson(4;6;TRUE)

And return the probability of 0.285057 = 28.5057%

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Using Excel

Try the poisson distribution using Excel… Download and open “Binomial Poisson Distribution”

Excel file… Try “Poisson”………

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Poisson Distribution Heritage TitlePoisson 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.

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Poisson Distribution – Heritage Tile

Enter values zero through 10

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Poisson Distribution – Heritage Tile

Select Formulas,

More Functions,

Statistical and

POISSON

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Poisson Distribution – Heritage Tile

Enter:

a1, 3, false

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Poisson Distribution – Heritage Tile

Notice that I had pre-selected Cell B1.

When I pressed enter the Poisson Probability was

loaded into that cell.

Simply copy and paste Cell B1 into cells B2 : B11

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Poisson Distribution – Heritage Tile

•Select the Insert tab

•Select Column

•Select the chart type that you want

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Poisson Distribution – Heritage Tile

Format the chart as per Chapter 2

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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