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2007 Pearson Education

Chapter 3: Random Variablesand Probability Distributions

Part 1: Probability and Probability

Distributions

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Experiments and Outcomes Experiment the observation of some activity

or the act of taking a measurement

Roll dice Measure dimension of a manufactured good

Conduct market survey

Outcome a particular result of anexperiment

Sum of the dice

Length of the dimension

Proportion of respondents that favor a product

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Events Sample space - all possible outcomes of an

experiment Dice rolls of 2, 3, , 12 All real numbers corresponding to a dimension A proportion between 0 and 1

An event is a collection of one or moreoutcomes from Obtaining a 7 or 11 on a roll of dice Having a manufactured dimension larger or

smaller than the required tolerance The proportion of respondents that favor a

product is at least 0.60

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Probability Probability a measure (between 0 and 1) of

the likelihood that an event will occur

Three views: Classical definition: based on theory

Relative frequency: based on empirical data

Subjective: based on judgment

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Classical Definition Probability = number of favorable outcomes

divided by the total number of possible

outcomes Example: There are six ways of rolling a 7

with a pair of dice, and 36 possible rolls.Therefore, the probability of rolling a 7 is

6/36 = .167.

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Relative Frequency Definition Probability = number of times an event has

occurred in the past divided by the total

number of observations Example: Of the last 10 days when certain

weather conditions have been observed, ithas rained the next day 8 times. The

probability of rain the next day is 0.80

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Subjective Definition What is the probability that the New York

Yankees will win the World Series this year?

What is the probability your school will win itsconference championship this year?

What is the probability the NASDAQ will goup 3% next week?

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Probability Rules1. Probability associated with any outcome must be

between 0 and 1

2. Sum of probabilities over all possible outcomesmust be 1.0 Example: Flip a coin three times

Outcomes: HHH, HHT, HTH, THH, HTT, THT, TTH,TTT

Each has probability of (1/2)3 = 1/8

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Mutually Exclusive Events Two events are mutually exclusive if the

occurrence of any one means that none

of the other can occur at the same time.

CB

B & C

A

B

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Probability Calculations1. Probability of any event is the sum of

the outcomes that compose that event

2. If events A and B are mutually exclusive,then

P(A or B) = P(A) + P(B)

3. If events A and B are not mutuallyexclusive, thenP(A or B) = P(A) + P(B) P(A and B)

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Example What is the probability of obtaining exactly

two heads or exactly two tails in 3 flips of acoin? These events are mutually exclusive. Probability =

3/8 + 3/8 = 6/8

What is the probability of obtaining at leasttwo tails or at least one head? A = {TTT, TTH, THT, HTT}, B = {TTH, THT, THH,

HTT, HTH, HHT, HHH} The events are notmutually exclusive.

P(A) = 4/8; P(B) = 7/8; P(A&B) = 3/8. Therefore,

P(A or B) = 4/8 + 7/8 3/8 = 1

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Conditional Probability Conditional probability the probability

of the occurrence of one event, given

that the other event is known to haveoccurred.

P(A|B) = P(A and B)/P(B)

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Example What is the probability of exactly 2 heads in

three flips, given that the first flip is a head?

HHH, HHT, HTH, THH, HTT, THT, TTH, TTT A = exactly 2 heads in 3 flips

B = first flip is a head, P(B) = 4/8

A&B = {HHT, HTH} P(A&B) = 2/8

P(A|B) = P(A&B)/P(B) = (2/8)/(4/8) = i.e.:

B ={HHH, HHT, HTH, HTT};A shown in red

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Joint Probability Joint probability table probabilities of

outcomes of two different variables

Variable 1: Number of heads in three flipsof a coin

Variable 2: First flip (H or T)

Frequencies Joint Probabilities

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Marginal Probability

P(A|B) = P(A&B) / P(B) = (1/4)/(1/2) =

Marginalprobabilities

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Other Uses for Conditional

Probability P(A and B) = P(B|A)*P(A)

P(A) = P(A|B1)*P(B1) + P(A|B2)*P(B2) +

+ P(A|Bk)*P(Bk)

where B1, B2, , Bkare k mutuallyexclusive and exhaustive outcomes.

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Statistical Independence Two events A and B are statistically

independent if the probability of A given

B equals the probability of A; in otherwords P(A|B) = P(A) or P(B|A) = P(B).

Implications for marketing

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Bayes Theorem Suppose that A1, A2, , Ak is a set of

mutually exclusive and collectively

exhaustive events, and seek theprobability that some event Ai occursgiven that another event B has occurred.

P(B|Ai)*P(Ai)

P(Ai|B) = --------------------------------------------------------------------

P(B|A1)*P(A1) + P(B|A2)*P(A2) + + P(B|Ak)*P(Ak)

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Example Suppose that a digital camera manufacturer is

developing a new model. Historically, 70 percent ofcameras that have been introduced have been

successful it terms of meeting a required return oninvestment, while 30 percent have been unsuccessful.Analysis of past market research studies, conductedprior to worldwide market introduction, has found that90 percent of all successful product introductions had

received favorable consumer response, while 60percent of all unsuccessful products had also receivedfavorable consumer response. If the new modelreceives a favorable response from a market researchstudy, what is the probability that it will actually be

successful?

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Example (continued) Events

A1 = new camera successful, P(A1) = 0.7

A2 = new camera unsuccessful, P(A2) = 0.3 B1 = marketing response favorable, B2 = marketing response unfavorable

Other know information: P(B1|A1) = 0.9 P(B1|A2) = 0.6

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Applying Bayes TheoremP(B1|A1)*P(A1)

P(A1|B1) = -------------------------------------------------------------------------

P(B1

|A1

)*P(A1

) + P(B1

|A2

)*P(A2

) + + P(B1

|Ak

)*P(Ak

)

(0.9)(0.7)

P(A1|B1) = -------------------------------- = 0.778

(0.9)(0.7) + (0.6)(0.3)

Although 70 percent of all previous new models have beensuccessful, knowing that the marketing report is favorableincreases the likelihood to 77.8 percent.

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Random Variables Random variable a function that assigns a real

number to each element of a sample space (anumerical description of the outcome of an

experiment).Random variables are denoted by capitalletters, X, Y, ; specific values by lower case letters,x, y,

Random variables may be discrete, with a finite or

infinitely countable number of values (number ofinaccurate orders or number of imperfections on acar) or continuous, having any real value possiblywithin some limited range(tomorrows temperature)

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Examples of Random Variables Experiment: flip a coin 3 times.

Outcomes: TTT, TTH, THT, THH, HTT, HTH, HHT,

HHH Random variable: X = number of heads.

X can be either 0, 1, 2, or 3.

Experiment: observe end-of-week closing

stock price. Random variable: Y = closing stock price.

X can be any nonnegative real number.

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Probability Distributions Probability distribution a characterization of

the possible values a random variable may

assume along with the probability ofoccurrence.

Probability distributions may be defined forboth discrete and continuous random

variables.

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Discrete Random Variables Probability mass functionf(x): specifies the

probability of each discrete outcome

Two properties: 0 f(xi) 1f(xi) = 1

Cumulative distribution function, F(x): specifiesthe probability that the random variable will beless than or equal tox.

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Example: Census Data

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Charts for f(x) and F(x)

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Example: Roll Two DiceSumofdice, x

2 3 4 5 6 7 8 9 10 11 12

f(x) 1/36 2/36 3/36 4/36 5/36 6/36 5/36 4/36 3/36 2/36 1/36

F(x) 1/36 3/36 6/36 10/36 15/36 21/36 26/36 30/36 33/36 35/36 1.0

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Joint Probability Distributions Joint probability distribution of two random variables,

f(x,y): specifies that probability that X = x and Y = ysimultaneously.

Not a High

School Grad

High School

Graduate

Some College

No Degree

Associate's

Degree

Bachelor's

Degree

Advanced

Degree

Age

25-34 0.027 0.073 0.045 0.020 0.049 0.014 0.228

35-44 0.031 0.088 0.047 0.024 0.047 0.021 0.258

45-54 0.026 0.063 0.035 0.017 0.035 0.022 0.198

55-64 0.026 0.048 0.019 0.007 0.017 0.012 0.129

65-74 0.030 0.038 0.014 0.004 0.010 0.007 0.104

75 and older 0.031 0.027 0.011 0.003 0.007 0.004 0.082

0.172 0.338 0.172 0.075 0.164 0.080 1.000

Joint Marginal

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Continuous Random Variables Probability density function, f(x), a continuous

function that describes the probability of

outcomes for the random variable X. Ahistogram of sample data approximates theshape of the underlying density function.

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Properties of Probability

Density Functions f(x) 0 for all x

Total area under f(x) = 1

There are always infinitely many values for X P(X = x) = 0

We can only define probabilities over

intervals: e.g.,P( c < X < d), P(X < c), or P(X > d)

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Cumulative Distribution

Function F(x) specifies the probability that the random

variable X will be less than or equal to x; that

is, P(X x). F(x) is equal to the area under f(x) to the left

of x

The probability that X is between a and b isthe area under f(x) from a to b = F(b) F(a)

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Example

f(x) = 4x 16 for 4 x 4.5

= -4x + 20 for 4.5 x 5

F(x) = 2x2 16x + 32 for 4 x 4.5

= -2x2 + 20x 49 for 4.5 x 5

P(X > 4.7) = F(5) F(4.7) = 1 - 0.82 = 0.18

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Cumulative Distribution Function

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Expected Value and Variance

of Random Variables Expected value of a random variable X is the theoretical

analogy of the mean, or weighted average of possiblevalues:

Variance and standard deviation of a random variable X:

1=i

ii )f(xx=E[X]

1j =

j2

j )f(xE[X])-(x=Var[X]

1j =

j

2

jX )f(xE[X])-(x=

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Example You play a lottery in which you buy a ticket

for $50 and are told you have a 1 in 1000

chance of winning $25,000. The randomvariable X is your net winnings, and itsprobability distribution is

x f(x)

-$50 0.999

$24,950 0.001

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Calculations E[X] = -$50(0.999) + $24,950(0.001) =

-$25.00

Var[X] = (-50 - [-25.00])2(0.999) +(24,950 - [-25.00])2(0.001) = 624,375

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Functions of Random

Variables Two useful formulas:

E[aX] = aE[X]

Var[aX] = a2Var[X]

where a is any constant

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Discrete Probability

Distributions Bernoulli

Binomial

Poisson

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Bernoulli Distribution A random variable with two possible

outcomes (x = 0 and x = 1), each with

constant probabilities of occurrence:

f(x) = p if x = 1

f(x) = 1 p if x = 0

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Binomial Distribution n independent replications of a Bernoulli trial

with constant probability of success p on eachtrial

Expected value = np

Variance = np(1-p)

x

n

= )!(!!

xnx

n

otherwise

nxforppxnxf xnx

0

,...,2,1,0)1()(

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Example If the probability that any individual will react

positively to a new drug is 0.8, what is the probabilitydistribution that 4 individuals will react positively out

of a sample of 10

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Excel Function BINOMDIST(number_s, trials, probability_s,

cumulative)

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Examples of the Binomial

Distribution

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Poisson Distribution Models the number of occurrences in some unit of

measure, e.g., events per unit time, number of itemsper order

X = number of events that occur; x = 0, 1, 2,

Expected value = l; variance = l

Poisson approximates binomial when n is large and psmall

otherwise

xforx

exf

x

0

,...2,1,0!

)(

ll

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Example Suppose that the average number of customers

arriving at an ATM during lunch hour is l = 12customers per hour. The probability that exactly x

customers will arrive during the hour is

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Excel Function POISSON (x, mean, cumulative)

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Poisson Distribution (l = 12)

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Properties of Continuous

Distributions Continuous distributions have one or

more parameters that characterize the

density function: Shape parameter controls the shape of

the distribution

Scale parameter controls the unit of

measurement Location parameter specifies the location

relative to zero on the horizontal axis

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Uniform Distribution Density function

Distribution function

f xb a

if a x b( )

1

0

1

if x a

F xx a

b aif a x b

if b x

( )

a b

x

f(x)m = (a + b)/22 = (b a)2/12

a = location

b

a = scale

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Normal Distribution Familiar bell-shaped curve.

Symmetric, median = mean = mode; half the area is oneither side of the mean

Range is unbounded: the curve never touches the x-axisParameters

Mean, m (location)

Variance 2 > 0 (scale)

Density function:

f(x) =e

-(x - )2

m

2

2

2

2

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Standard Normal Distribution Standard normal: mean = 0, variance = 1,

denoted as N(0,1)

See Appendix Table A.1

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Standardized Normal Values Transformation from N(m,) to N(0,1):

Standardized z-values are expressed in units

of standard deviations of X.

mx=z

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Areas Under the Normal

Density About 68.3% is within one sigma of the mean

About 95.4% is within two sigma of the mean

About 99.7% is within three sigma of themean

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Normal Probability

Calculations Customer demand averages 750

units/month with a standard deviation of

100 units/month. Find P(X>900), P(X>700), P(700

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P(X>900)5.1

100

750900=z

Area = 0.9332

Area = 0.0668

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P(X>700)

5.0100

750700=z

P(X > 700) = P(Z > -0.5) = 1 - P(Z < -0.5).From Table A.1, P(Z < -0.5) = 0.3085.

Thus, P(X > 700) = 1 - 0.3085 = 0.6915.

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P(700

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P(X>x)>.10

28.1100

750=z

xFirst, find the valueof z from Table A.1,then solve for x

x = 878

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Excel Support

NORMDIST(x, mean, standard_deviation,cumulative)

NORMSDIST(z): same as Table A.1 STANDARDIZE(x, mean, standard_deviation)

computes z-values

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Computing Normal Probabilities

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Triangular Distribution

Three parameters:

Minimum, a

Maximum, b

Most likely, c

a is the location parameter;

(b a) the scale parameter,

c the shape parameter.

2

2

0

( )

( )( )

( )( )

( )( )

x a

b a c aif a x c

f xb x

b a b cif c x b

otherwise

Mean = (a + b + c)/3

Variance = (a2

+ b2

+ c2

ab

ac

bc)/18

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Exponential Distribution

Models events that occur randomly over time

Customer arrivals, machine failures

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Properties of Exponential

Density function

Distribution function

Mean = 1/l Variance = 1/l2

Excel function EXPONDIST(x, lambda, cumulative).

f(x) = le-lx, x 0

F(x) = 1 - e-lx, x 0

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Lognormal Distribution

X is lognormal if the distribution of lnX isnormal

Positively skewed, bounded below by 0 Models task times, stock and real estate

prices

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Gamma Distribution

Family of distributions defined by shape a,scale b, and location L

Defined for x > L Models task times, time between events,

inventories

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Weibull Distribution

Family of distributions defined by shape a,scale b, and location L

When L = 0 and b = 1, same as exponentialwith l = 1/a.

Models results from life tests, equipmentfailures, and task times.

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Beta Distribution

Defined over the range (0, s)

Two shape parameters a and b; if equal, beta

is symmetric; a < b, positively skewed; ifeither equals 1 and the other > 1, Jshaped.

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Geometric Distribution

Sequence of Bernoulli trials that describes thenumber of trials until the first success.

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Negative Binomial

Distribution of number of trials until the rthsuccess.

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Logistic Distribution

Describes the growth of a population overtime.

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Pareto Distribution

Distribution in which a small proportion ofitems accounts for a large proportion of somecharacteristic.

PHSt t T l P b bilit &

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PHStatTool: Probability &Probability Distributions

PHStatmenu > Probability & ProbabilityDistributions> choice of distribution:

Normal

Binomial

Exponential

Poisson

Hypergeometric

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PHStatOutput