Chap 5-1 A Course In Business Statistics, 4th © 2006 Prentice-Hall, Inc. A Course In Business...

Chap 5-1A Course In Business Statistics, 4th © 2006 Prentice-Hall, Inc.

A Course In Business Statistics

4th Edition

Chapter 5Discrete and Continuous Probability Distributions

Chap 5-2A Course In Business Statistics, 4th © 2006 Prentice-Hall, Inc.

Chapter GoalsAfter completing this chapter, you should be

able to: Apply the binomial distribution to applied problems Compute probabilities for the Poisson and

hypergeometric distributions Find probabilities using a normal distribution table

and apply the normal distribution to business problems

Recognize when to apply the uniform and exponential distributions

Chap 5-3A Course In Business Statistics, 4th © 2006 Prentice-Hall, Inc.

Probability Distributions

Continuous Probability

Distributions

Binomial

Hypergeometric

Poisson

Probability Distributions

Discrete Probability

Distributions

Normal

Uniform

Exponential

Chap 5-4A Course In Business Statistics, 4th © 2006 Prentice-Hall, Inc.

A discrete random variable is a variable that can assume only a countable number of valuesMany possible outcomes: number of complaints per day number of TV’s in a household number of rings before the phone is answered

Only two possible outcomes: gender: male or female defective: yes or no spreads peanut butter first vs. spreads jelly first

Discrete Probability Distributions

Chap 5-5A Course In Business Statistics, 4th © 2006 Prentice-Hall, Inc.

Continuous Probability Distributions

A continuous random variable is a variable that can assume any value on a continuum (can assume an uncountable number of values) thickness of an item time required to complete a task temperature of a solution height, in inches

These can potentially take on any value, depending only on the ability to measure accurately.

Chap 5-6A Course In Business Statistics, 4th © 2006 Prentice-Hall, Inc.

The Binomial Distribution

Binomial

Hypergeometric

Poisson

Probability Distributions

Discrete Probability

Distributions

Chap 5-7A Course In Business Statistics, 4th © 2006 Prentice-Hall, Inc.

The Binomial Distribution Characteristics of the Binomial Distribution:

A trial has only two possible outcomes – “success” or “failure”

There is a fixed number, n, of identical trials The trials of the experiment are independent of each

other The probability of a success, p, remains constant from

trial to trial If p represents the probability of a success, then

(1-p) = q is the probability of a failure

Chap 5-8A Course In Business Statistics, 4th © 2006 Prentice-Hall, Inc.

Binomial Distribution Settings

A manufacturing plant labels items as either defective or acceptable

A firm bidding for a contract will either get the contract or not

A marketing research firm receives survey responses of “yes I will buy” or “no I will not”

New job applicants either accept the offer or reject it

Chap 5-9A Course In Business Statistics, 4th © 2006 Prentice-Hall, Inc.

Counting Rule for Combinations

A combination is an outcome of an experiment where x objects are selected from a group of n objects

)!xn(!x!nCn

x

where:n! =n(n - 1)(n - 2) . . . (2)(1)

x! = x(x - 1)(x - 2) . . . (2)(1) 0! = 1 (by definition)

Chap 5-10A Course In Business Statistics, 4th © 2006 Prentice-Hall, Inc.

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!

( )!

Example: Flip a coin four times, let x = # heads:

n = 4

p = 0.5

q = (1 - .5) = .5

x = 0, 1, 2, 3, 4

Binomial Distribution Formula

Chap 5-11A Course In Business Statistics, 4th © 2006 Prentice-Hall, Inc.

n = 5 p = 0.1

n = 5 p = 0.5

Mean

0.2.4.6

0 1 2 3 4 5X

P(X)

.2

.4

.6

0 1 2 3 4 5X

P(X)

0

Binomial Distribution The shape of the binomial distribution depends on the

values of p and n

Here, n = 5 and p = .1

Here, n = 5 and p = .5

Chap 5-12A Course In Business Statistics, 4th © 2006 Prentice-Hall, Inc.

Binomial Distribution Characteristics

Mean

Variance and Standard Deviation

npE(x)μ

npqσ2

npqσ

Where n = sample sizep = probability of successq = (1 – p) = probability of failure

Chap 5-13A Course In Business Statistics, 4th © 2006 Prentice-Hall, Inc.

n = 5 p = 0.1

n = 5 p = 0.5

Mean

0.2.4.6

0 1 2 3 4 5X

P(X)

.2

.4

.6

0 1 2 3 4 5X

P(X)

0

0.5(5)(.1)npμ

0.6708

.1)(5)(.1)(1npqσ

2.5(5)(.5)npμ

1.118

.5)(5)(.5)(1npqσ

Binomial CharacteristicsExamples

Chap 5-14A Course In Business Statistics, 4th © 2006 Prentice-Hall, Inc.

Using Binomial Tablesn = 10

x p=.15 p=.20 p=.25 p=.30 p=.35 p=.40 p=.45 p=.50

012345678910

0.19690.34740.27590.12980.04010.00850.00120.00010.00000.00000.0000

0.10740.26840.30200.20130.08810.02640.00550.00080.00010.00000.0000

0.05630.18770.28160.25030.14600.05840.01620.00310.00040.00000.0000

0.02820.12110.23350.26680.20010.10290.03680.00900.00140.00010.0000

0.01350.07250.17570.25220.23770.15360.06890.02120.00430.00050.0000

0.00600.04030.12090.21500.25080.20070.11150.04250.01060.00160.0001

0.00250.02070.07630.16650.23840.23400.15960.07460.02290.00420.0003

0.00100.00980.04390.11720.20510.24610.20510.11720.04390.00980.0010

109876543210

p=.85 p=.80 p=.75 p=.70 p=.65 p=.60 p=.55 p=.50 x

Examples: n = 10, p = .35, x = 3: P(x = 3|n =10, p = .35) = .2522

n = 10, p = .75, x = 2: P(x = 2|n =10, p = .75) = .0004

Chap 5-15A Course In Business Statistics, 4th © 2006 Prentice-Hall, Inc.

Using PHStat Select PHStat / Probability & Prob. Distributions / Binomial…

Chap 5-16A Course In Business Statistics, 4th © 2006 Prentice-Hall, Inc.

Using PHStat Enter desired values in dialog box

Here: n = 10p = .35

Output for x = 0 to x = 10 will be generated by PHStat

Optional check boxesfor additional output

Chap 5-17A Course In Business Statistics, 4th © 2006 Prentice-Hall, Inc.

P(x = 3 | n = 10, p = .35) = .2522

PHStat Output

P(x > 5 | n = 10, p = .35) = .0949

Chap 5-18A Course In Business Statistics, 4th © 2006 Prentice-Hall, Inc.

The Poisson Distribution

Binomial

Hypergeometric

Poisson

Probability Distributions

Discrete Probability

Distributions

Chap 5-19A Course In Business Statistics, 4th © 2006 Prentice-Hall, Inc.

The Poisson Distribution

Characteristics of the Poisson Distribution: The outcomes of interest are rare relative to the

possible outcomes The average number of outcomes of interest per time

or space interval is The number of outcomes of interest are random, and

the occurrence of one outcome does not influence the chances of another outcome of interest

The probability of that an outcome of interest occurs in a given segment is the same for all segments

Chap 5-20A Course In Business Statistics, 4th © 2006 Prentice-Hall, Inc.

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 sizee = base of the natural logarithm system (2.71828...)

!xe)t()x(P

tx

Chap 5-21A Course In Business Statistics, 4th © 2006 Prentice-Hall, Inc.

Poisson Distribution Characteristics

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

Chap 5-22A Course In Business Statistics, 4th © 2006 Prentice-Hall, Inc.

Using Poisson Tables

X

t

0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90

01234567

0.90480.09050.00450.00020.00000.00000.00000.0000

0.81870.16370.01640.00110.00010.00000.00000.0000

0.74080.22220.03330.00330.00030.00000.00000.0000

0.67030.26810.05360.00720.00070.00010.00000.0000

0.60650.30330.07580.01260.00160.00020.00000.0000

0.54880.32930.09880.01980.00300.00040.00000.0000

0.49660.34760.12170.02840.00500.00070.00010.0000

0.44930.35950.14380.03830.00770.00120.00020.0000

0.40660.36590.16470.04940.01110.00200.00030.0000

Example: Find P(x = 2) if = .05 and t = 100

.07582!

e(0.50)!xe)t()2x(P

0.502tx

Chap 5-23A Course In Business Statistics, 4th © 2006 Prentice-Hall, Inc.

Graph of Poisson Probabilities

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0 1 2 3 4 5 6 7

x

P(x)X

t =0.50

01234567

0.60650.30330.07580.01260.00160.00020.00000.0000

P(x = 2) = .0758

Graphically: = .05 and t = 100

Chap 5-24A Course In Business Statistics, 4th © 2006 Prentice-Hall, Inc.

Poisson Distribution Shape The shape of the Poisson Distribution

depends on the parameters and t:

0.00

0.05

0.10

0.15

0.20

0.25

1 2 3 4 5 6 7 8 9 10 11 12

x

P(x

)

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0 1 2 3 4 5 6 7

x

P(x)

t = 0.50 t = 3.0

Chap 5-25A Course In Business Statistics, 4th © 2006 Prentice-Hall, Inc.

The Hypergeometric Distribution

Binomial

Poisson

Probability Distributions

Discrete Probability

Distributions

Hypergeometric

Chap 5-26A Course In Business Statistics, 4th © 2006 Prentice-Hall, Inc.

The Hypergeometric Distribution

“n” trials in a sample taken from a finite population of size N

Sample taken without replacement Trials are dependent Concerned with finding the probability of “x”

successes in the sample where there are “X” successes in the population

Chap 5-27A Course In Business Statistics, 4th © 2006 Prentice-Hall, Inc.

Hypergeometric Distribution Formula

Nn

Xx

XNxn

CCC)x(P

.

WhereN = Population sizeX = number of successes in the populationn = sample sizex = number of successes in the sample

n – x = number of failures in the sample

(Two possible outcomes per trial)

Chap 5-28A Course In Business Statistics, 4th © 2006 Prentice-Hall, Inc.

Hypergeometric Distribution Formula

0.3120

(6)(6)C

CCC

CC2)P(x 103

42

61

Nn

Xx

XNxn

■ Example: 3 Light bulbs were selected from 10. Of the 10 there were 4 defective. What is the probability that 2 of the 3 selected are defective?

N = 10 n = 3 X = 4 x = 2

Chap 5-29A Course In Business Statistics, 4th © 2006 Prentice-Hall, Inc.

Hypergeometric Distribution in PHStat

Select:PHStat / Probability & Prob. Distributions / Hypergeometric …

Chap 5-30A Course In Business Statistics, 4th © 2006 Prentice-Hall, Inc.

Hypergeometric Distribution in PHStat

Complete dialog box entries and get output …

N = 10 n = 3X = 4 x = 2

P(x = 2) = 0.3

(continued)

Chap 5-31A Course In Business Statistics, 4th © 2006 Prentice-Hall, Inc.

The Normal Distribution

Continuous Probability

Distributions

Probability Distributions

Normal

Uniform

Exponential

Chap 5-32A Course In Business Statistics, 4th © 2006 Prentice-Hall, Inc.

The Normal Distribution ‘Bell Shaped’ Symmetrical Mean, Median and Mode

are EqualLocation is determined by the mean, μSpread is determined by the standard deviation, σ

The random variable has an infinite theoretical range: + to

Mean = Median = Mode

x

f(x)

μ

σ

Chap 5-33A Course In Business Statistics, 4th © 2006 Prentice-Hall, Inc.

By varying the parameters μ and σ, we obtain different normal distributions

Many Normal Distributions

Chap 5-34A Course In Business Statistics, 4th © 2006 Prentice-Hall, Inc.

The Normal Distribution Shape

x

f(x)

μ

σ

Changing μ shifts the distribution left or right.

Changing σ increases or decreases the spread.

Chap 5-35A Course In Business Statistics, 4th © 2006 Prentice-Hall, Inc.

Finding Normal Probabilities

Probability is the area under thecurve!

a b x

f(x) P a x b( )

Probability is measured by the area under the curve

Chap 5-36A Course In Business Statistics, 4th © 2006 Prentice-Hall, Inc.

f(x)

xμ

Probability as Area Under the Curve

0.50.5

The total area under the curve is 1.0, and the curve is symmetric, so half is above the mean, half is below

1.0)xP(

0.5)xP(μ 0.5μ)xP(

Chap 5-37A Course In Business Statistics, 4th © 2006 Prentice-Hall, Inc.

Empirical Rules

μ ± 1σencloses about 68% of x’s

f(x)

xμ μσμσ

What can we say about the distribution of values around the mean? There are some general rules:

σσ

68.26%

Chap 5-38A Course In Business Statistics, 4th © 2006 Prentice-Hall, Inc.

The Empirical Rule μ ± 2σ covers about 95% of x’s μ ± 3σ covers about 99.7% of x’s

xμ2σ 2σ

xμ3σ 3σ

95.44% 99.72%

(continued)

Chap 5-39A Course In Business Statistics, 4th © 2006 Prentice-Hall, Inc.

Importance of the Rule If a value is about 2 or more standard deviations away from the mean in a normal distribution, then it is far from the mean

The chance that a value that far or farther away from the mean is highly unlikely, given that particular mean and standard deviation

Chap 5-40A Course In Business Statistics, 4th © 2006 Prentice-Hall, Inc.

The Standard Normal Distribution

Also known as the “z” distribution Mean is defined to be 0 Standard Deviation is 1

z

f(z)

0

1

Values above the mean have positive z-values, values below the mean have negative z-values

Chap 5-41A Course In Business Statistics, 4th © 2006 Prentice-Hall, Inc.

The Standard Normal

Any normal distribution (with any mean and standard deviation combination) can be transformed into the standard normal distribution (z)

Need to transform x units into z units

Chap 5-42A Course In Business Statistics, 4th © 2006 Prentice-Hall, Inc.

Translation to the Standard Normal Distribution

Translate from x to the standard normal (the “z” distribution) by subtracting the mean of x and dividing by its standard deviation:

σμxz

Chap 5-43A Course In Business Statistics, 4th © 2006 Prentice-Hall, Inc.

Example If x is distributed normally with mean of 100

and standard deviation of 50, the z value for x = 250 is

This says that x = 250 is three standard deviations (3 increments of 50 units) above the mean of 100.

3.050

100250σ

μxz

Chap 5-44A Course In Business Statistics, 4th © 2006 Prentice-Hall, Inc.

Comparing x and z units

z100

3.00250 x

Note that the distribution is the same, only the scale has changed. We can express the problem in original units (x) or in standardized units (z)

μ = 100

σ = 50

Chap 5-45A Course In Business Statistics, 4th © 2006 Prentice-Hall, Inc.

The Standard Normal Table

The Standard Normal table in the textbook (Appendix D)

gives the probability from the mean (zero) up to a desired value for z

z0 2.00

.4772Example: P(0 < z < 2.00) = .4772

Chap 5-46A Course In Business Statistics, 4th © 2006 Prentice-Hall, Inc.

The Standard Normal Table

The value within the table gives the probability from z = 0 up to the desired z value

z 0.00 0.01 0.02 …

0.1

0.2

.4772

2.0P(0 < z < 2.00) = .4772

The row shows the value of z to the first decimal point

The column gives the value of z to the second decimal point

2.0

.

.

.

(continued)

Chap 5-47A Course In Business Statistics, 4th © 2006 Prentice-Hall, Inc.

General Procedure for Finding Probabilities

Draw the normal curve for the problem in terms of x

Translate x-values to z-values

Use the Standard Normal Table

To find P(a < x < b) when x is distributed normally:

Chap 5-48A Course In Business Statistics, 4th © 2006 Prentice-Hall, Inc.

Z Table example Suppose x is normal with mean 8.0 and

standard deviation 5.0. Find P(8 < x < 8.6)

P(8 < x < 8.6)

= P(0 < z < 0.12)

Z0.12 0x8.6 8

05

88σ

μxz

0.125

88.6σ

μxz

Calculate z-values:

Chap 5-49A Course In Business Statistics, 4th © 2006 Prentice-Hall, Inc.

Z Table example Suppose x is normal with mean 8.0 and

standard deviation 5.0. Find P(8 < x < 8.6)

P(0 < z < 0.12)

z0.12 0x8.6 8

P(8 < x < 8.6)

= 8 = 5

= 0 = 1

(continued)

Chap 5-50A Course In Business Statistics, 4th © 2006 Prentice-Hall, Inc.

Z

0.12

z .00 .01

0.0 .0000 .0040 .0080

.0398 .0438

0.2 .0793 .0832 .0871

0.3 .1179 .1217 .1255

Solution: Finding P(0 < z < 0.12)

.0478.02

0.1 .0478

Standard Normal Probability Table (Portion)

0.00

= P(0 < z < 0.12)P(8 < x < 8.6)

Chap 5-51A Course In Business Statistics, 4th © 2006 Prentice-Hall, Inc.

Finding Normal Probabilities Suppose x is normal with mean 8.0

and standard deviation 5.0. Now Find P(x < 8.6)

Z

8.68.0

Chap 5-52A Course In Business Statistics, 4th © 2006 Prentice-Hall, Inc.

Finding Normal Probabilities Suppose x is normal with mean 8.0

and standard deviation 5.0. Now Find P(x < 8.6)

(continued)

Z

0.12

.0478

0.00

.5000 P(x < 8.6)

= P(z < 0.12)

= P(z < 0) + P(0 < z < 0.12)

= .5 + .0478 = .5478

Chap 5-53A Course In Business Statistics, 4th © 2006 Prentice-Hall, Inc.

Upper Tail Probabilities Suppose x is normal with mean 8.0

and standard deviation 5.0. Now Find P(x > 8.6)

Z

8.68.0

Chap 5-54A Course In Business Statistics, 4th © 2006 Prentice-Hall, Inc.

Now Find P(x > 8.6)…(continued)

Z

0.12 0

Z

0.12

.0478

0

.5000 .50 - .0478 = .4522

P(x > 8.6) = P(z > 0.12) = P(z > 0) - P(0 < z < 0.12)

= .5 - .0478 = .4522

Upper Tail Probabilities

Chap 5-55A Course In Business Statistics, 4th © 2006 Prentice-Hall, Inc.

Lower Tail Probabilities Suppose x is normal with mean 8.0

and standard deviation 5.0. Now Find P(7.4 < x < 8)

Z

7.48.0

Chap 5-56A Course In Business Statistics, 4th © 2006 Prentice-Hall, Inc.

Lower Tail Probabilities Now Find P(7.4 < x < 8)…

Z

7.48.0

The Normal distribution is symmetric, so we use the same table even if z-values are negative:

P(7.4 < x < 8)

= P(-0.12 < z < 0)

= .0478

(continued)

.0478

Chap 5-57A Course In Business Statistics, 4th © 2006 Prentice-Hall, Inc.

Normal Probabilities in PHStat

We can use Excel and PHStat to quickly generate probabilities for any normal distribution

We will find P(8 < x < 8.6) when x is normally distributed with mean 8 and standard deviation 5

Chap 5-58A Course In Business Statistics, 4th © 2006 Prentice-Hall, Inc.

PHStat Dialogue Box

Select desired options and enter values

Chap 5-59A Course In Business Statistics, 4th © 2006 Prentice-Hall, Inc.

PHStat Output

Chap 5-60A Course In Business Statistics, 4th © 2006 Prentice-Hall, Inc.

The Uniform Distribution

Continuous Probability

Distributions

Probability Distributions

Normal

Uniform

Exponential

Chap 5-61A Course In Business Statistics, 4th © 2006 Prentice-Hall, Inc.

The Uniform Distribution

The uniform distribution is a probability distribution that has equal probabilities for all possible outcomes of the random variable

Chap 5-62A Course In Business Statistics, 4th © 2006 Prentice-Hall, Inc.

The Continuous Uniform Distribution:

otherwise 0

bxaifab

1

wheref(x) = value of the density function at any x valuea = lower limit of the intervalb = upper limit of the interval

The Uniform Distribution(continued)

f(x) =

Chap 5-63A Course In Business Statistics, 4th © 2006 Prentice-Hall, Inc.

Uniform DistributionExample: Uniform Probability Distribution

Over the range 2 ≤ x ≤ 6:

2 6

.25

f(x) = = .25 for 2 ≤ x ≤ 66 - 21

x

f(x)

Chap 5-64A Course In Business Statistics, 4th © 2006 Prentice-Hall, Inc.

The Exponential Distribution

Continuous Probability

Distributions

Probability Distributions

Normal

Uniform

Exponential

Chap 5-65A Course In Business Statistics, 4th © 2006 Prentice-Hall, Inc.

The Exponential Distribution

Used to measure the time that elapses between two occurrences of an event (the time between arrivals)

Examples: Time between trucks arriving at an unloading

dock Time between transactions at an ATM Machine Time between phone calls to the main operator

Chap 5-66A Course In Business Statistics, 4th © 2006 Prentice-Hall, Inc.

The Exponential Distribution

aλe1a)xP(0

The probability that an arrival time is equal to or less than some specified time a is

where 1/ is the mean time between events

Note that if the number of occurrences per time period is Poisson with mean , then the time between occurrences is exponential with mean time 1/

Chap 5-67A Course In Business Statistics, 4th © 2006 Prentice-Hall, Inc.

Exponential Distribution Shape of the exponential distribution

(continued)

f(x)

x

= 1.0(mean = 1.0)

= 0.5 (mean = 2.0)

= 3.0(mean = .333)

Chap 5-68A Course In Business Statistics, 4th © 2006 Prentice-Hall, Inc.

Example

Example: Customers arrive at the claims counter at the rate of 15 per hour (Poisson distributed). What is the probability that the arrival time between consecutive customers is less than five minutes?

Time between arrivals is exponentially distributed with mean time between arrivals of 4 minutes (15 per 60 minutes, on average)

1/ = 4.0, so = .25

P(x < 5) = 1 - e-a = 1 – e-(.25)(5) = .7135

Chap 5-69A Course In Business Statistics, 4th © 2006 Prentice-Hall, Inc.

Chapter Summary

Reviewed key discrete distributions binomial, poisson, hypergeometric

Reviewed key continuous distributions normal, uniform, exponential

Found probabilities using formulas and tables

Recognized when to apply different distributions

Applied distributions to decision problems