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Transcript of 1 1 Slide Chapter 3-PART I Chapter 3-PART I John Kooti QUANTITATIVE METHODS FOR BUSINESS DISCREETE...
1 1 Slide Slide
Chapter 3-PART IChapter 3-PART I Chapter 3-PART IChapter 3-PART I
John KootiJohn Kooti
QUANTITATIVE QUANTITATIVE METHODS FORMETHODS FOR
BUSINESSBUSINESSDISCREETE PROBABILITY DISCREETE PROBABILITY
DISTRIBUTIONDISTRIBUTION
QUANTITATIVE QUANTITATIVE METHODS FORMETHODS FOR
BUSINESSBUSINESSDISCREETE PROBABILITY DISCREETE PROBABILITY
DISTRIBUTIONDISTRIBUTION
2 2 Slide Slide
Chapter 3, Part IChapter 3, Part I Discrete Probability Distributions Discrete Probability Distributions
Random VariablesRandom Variables Discrete Random VariablesDiscrete Random Variables Expected Value and VarianceExpected Value and Variance Binomial Probability DistributionBinomial Probability Distribution Poisson Probability DistributionPoisson Probability Distribution
3 3 Slide Slide
Random VariablesRandom Variables
A A random variablerandom variable is a numerical description is a numerical description of the outcome of an experiment.of the outcome of an experiment.
A random variable can be classified as being A random variable can be classified as being either discrete or continuous depending on the either discrete or continuous depending on the numerical values it assumes.numerical values it assumes.
A A discrete random variablediscrete random variable may assume either may assume either a finite number of values or an infinite a finite number of values or an infinite sequence of values.sequence of values.
A A continuous random variablecontinuous random variable may assume may assume any numerical value in an interval or collection any numerical value in an interval or collection of intervals.of intervals.
4 4 Slide Slide
Example: JSL AppliancesExample: JSL Appliances
Discrete random variable with a finite number of Discrete random variable with a finite number of values:values:
Let Let xx = number of TV sets sold at the store in one day = number of TV sets sold at the store in one day
where where xx can take on 5 values (0, 1, 2, 3, 4) can take on 5 values (0, 1, 2, 3, 4)
Discrete random variable with an infinite sequence of Discrete random variable with an infinite sequence of values:values:
Let Let xx = number of customers arriving in one day = number of customers arriving in one day
where where xx can take on the values 0, 1, 2, . . . can take on the values 0, 1, 2, . . .
We can count the customers arriving, but there is no We can count the customers arriving, but there is no finite upper limit on the number that might arrive.finite upper limit on the number that might arrive.
5 5 Slide Slide
Discrete Probability DistributionsDiscrete Probability Distributions
The The probability distributionprobability distribution for a random variable for a random variable describes how probabilities are distributed over the describes how probabilities are distributed over the values of the random variable.values of the random variable.
The probability distribution is defined by a The probability distribution is defined by a probabilityprobability functionfunction, denoted by , denoted by ff((xx), which provides the ), which provides the probability for each value of the random variable.probability for each value of the random variable.
The required conditions for a discrete probability The required conditions for a discrete probability function are:function are:
ff((xx) ) >> 0 0
ff((xx) = 1) = 1 We can describe a discrete probability distribution We can describe a discrete probability distribution
with a table, graph, or equation.with a table, graph, or equation.
6 6 Slide Slide
Example: JSL AppliancesExample: JSL Appliances
Using past data on TV sales (below left), a tabular Using past data on TV sales (below left), a tabular representation of the probability distribution for TV representation of the probability distribution for TV sales (below right) was developed.sales (below right) was developed.
NumberNumber
Units SoldUnits Sold of Daysof Days xx ff((xx))
00 8080 0 0 .40 .40
11 5050 1 1 .25 .25
22 4040 2 2 .20 .20
33 1010 3 3 .05 .05
44 2020 4 4 .10 .10
200200 1.00 1.00
7 7 Slide Slide
Example: JSL AppliancesExample: JSL Appliances
A graphical representation of the probability A graphical representation of the probability distribution for TV sales in one daydistribution for TV sales in one day
.10.10
.20.20
.30.30
.40.40
.50.50
0 1 2 3 40 1 2 3 4Values of Random Variable x (TV sales)Values of Random Variable x (TV sales)
Pro
bab
ilit
yP
rob
ab
ilit
y
8 8 Slide Slide
Expected Value and VarianceExpected Value and Variance
The The expected valueexpected value, or mean, of a random , or mean, of a random variable is a measure of its central location.variable is a measure of its central location.• Expected value of a discrete random Expected value of a discrete random
variable:variable:
EE((xx) = ) = = = xfxf((xx)) The The variancevariance summarizes the variability in the summarizes the variability in the
values of a random variable.values of a random variable.• Variance of a discrete random variable:Variance of a discrete random variable:
Var(Var(xx) = ) = 22 = = ((xx - - ))22ff((xx)) The The standard deviationstandard deviation, , , is defined as the , is defined as the
positive square root of the variance.positive square root of the variance.
9 9 Slide Slide
Example: JSL AppliancesExample: JSL Appliances
Expected Value of a Discrete Random VariableExpected Value of a Discrete Random Variable
xx ff((xx)) xfxf((xx))
00 .40.40 .00 .00
11 .25.25 .25 .25
22 .20.20 .40 .40
33 .05.05 .15 .15
44 .10.10 .40.40
1.20 = 1.20 = EE((xx))
The expected number of TV sets sold in a day The expected number of TV sets sold in a day is 1.2is 1.2
10 10 Slide Slide
Variance and Standard DeviationVariance and Standard Deviation
of a Discrete Random Variableof a Discrete Random Variable
xx x - x - ( (x - x - ))22 ff((xx)) ((xx - - ))22ff((xx)) _____ _________ ___________ _______ ____________________ _________ ___________ _______ _______________
00 -1.2-1.2 1.44 1.44 .40.40 .576 .576
11 -0.2-0.2 0.04 0.04 .25.25 .010 .010
22 0.8 0.8 0.64 0.64 .20.20 .128 .128
33 1.8 1.8 3.24 3.24 .05.05 .162 .162
44 2.8 2.8 7.84 7.84 .10.10 .784 .784
1.660 = 1.660 =
The variance of daily sales is 1.66 TV sets The variance of daily sales is 1.66 TV sets squaredsquared..
The standard deviation of sales is 1.29 TV sets.The standard deviation of sales is 1.29 TV sets.
Example: JSL AppliancesExample: JSL Appliances
11 11 Slide Slide
Example: JSL AppliancesExample: JSL Appliances
Formula Spreadsheet for Computing Expected Formula Spreadsheet for Computing Expected Value and VarianceValue and Variance
A B C D
1 x f(x) xf(x) (x- ) 2f(x)2 0 0.40 =A2*B2 =(A2-C$7) 2̂*B23 1 0.25 =A3*B3 =(A3-C$7) 2̂*B34 2 0.20 =A4*B4 =(A4-C$7) 2̂*B45 3 0.05 =A5*B5 =(A5-C$7) 2̂*B56 4 0.10 =A6*B6 =(A6-C$7) 2̂*B67 =SUM(C2:C6) =SUM(D2:D6)8 Expected Value Variance
12 12 Slide Slide
Binomial Probability DistributionBinomial Probability Distribution
Properties of a Binomial ExperimentProperties of a Binomial Experiment• The experiment consists of a sequence of The experiment consists of a sequence of nn
identical trials.identical trials.• Two outcomes, Two outcomes, successsuccess and and failurefailure, are , are
possible on each trial. possible on each trial. • The probability of a success, denoted by The probability of a success, denoted by pp, ,
does not change from trial to trial.does not change from trial to trial.• The trials are independent.The trials are independent.
13 13 Slide Slide
Example: Evans ElectronicsExample: Evans Electronics
Binomial Probability DistributionBinomial Probability Distribution
Evans is concerned about a low retention Evans is concerned about a low retention rate for employees. On the basis of past rate for employees. On the basis of past experience, management has seen a turnover of experience, management has seen a turnover of 10% of the hourly employees annually. Thus, for 10% of the hourly employees annually. Thus, for any hourly employees chosen at random, any hourly employees chosen at random, management estimates a probability of 0.1 that management estimates a probability of 0.1 that the person will not be with the company next year.the person will not be with the company next year.
Choosing 3 hourly employees a random, Choosing 3 hourly employees a random, what is the probability that 1 of them will leave what is the probability that 1 of them will leave the company this year?the company this year?
LetLet: : pp = .10, = .10, nn = 3, = 3, xx = 1 = 1
14 14 Slide Slide
Binomial Probability DistributionBinomial Probability Distribution
Binomial Probability FunctionBinomial Probability Function
wherewhere
ff((xx) = the probability of ) = the probability of xx successes in successes in nn trialstrials
nn = the number of trials = the number of trials
pp = the probability of success on any one = the probability of success on any one trialtrial
f xn
x n xp px n x( )
!!( )!
( )( )
1f xn
x n xp px n x( )
!!( )!
( )( )
1
15 15 Slide Slide
Example: Evans ElectronicsExample: Evans Electronics
Using the Binomial Probability FunctionUsing the Binomial Probability Function
= (3)(0.1)(0.81)= (3)(0.1)(0.81)
= .243 = .243
f xn
x n xp px n x( )
!!( )!
( )( )
1f xn
x n xp px n x( )
!!( )!
( )( )
1
f ( )!
!( )!( . ) ( . )1
31 3 1
0 1 0 91 2
f ( )!
!( )!( . ) ( . )1
31 3 1
0 1 0 91 2
16 16 Slide Slide
Example: Evans ElectronicsExample: Evans Electronics
Using the Tables of Binomial Probabilities Using the Tables of Binomial Probabilities
pn x .10 .15 .20 .25 .30 .35 .40 .45 .503 0 .7290 .6141 .5120 .4219 .3430 .2746 .2160 .1664 .1250
1 .2430 .3251 .3840 .4219 .4410 .4436 .4320 .4084 .37502 .0270 .0574 .0960 .1406 .1890 .2389 .2880 .3341 .37503 .0010 .0034 .0080 .0156 .0270 .0429 .0640 .0911 .1250
pn x .10 .15 .20 .25 .30 .35 .40 .45 .503 0 .7290 .6141 .5120 .4219 .3430 .2746 .2160 .1664 .1250
1 .2430 .3251 .3840 .4219 .4410 .4436 .4320 .4084 .37502 .0270 .0574 .0960 .1406 .1890 .2389 .2880 .3341 .37503 .0010 .0034 .0080 .0156 .0270 .0429 .0640 .0911 .1250
17 17 Slide Slide
Using a Tree DiagramUsing a Tree Diagram
FirstWorker
FirstWorker
Second
Worker
Second
Worker
ThirdWorke
r
ThirdWorke
r
Leaves (.1)Leaves (.1)
Stays (.9)Stays (.9)
Valueof x
Valueof x
33
22
00
22
22
11
Leaves (.1)Leaves (.1)
Leaves (.1)Leaves (.1)
S (.9)S (.9)
Stays (.9)Stays (.9)
Stays (.9)Stays (.9)
S (.9)S (.9)
S (.9)S (.9)
S (.9)S (.9)
L (.1)L (.1)
L (.1)L (.1)
L (.1)L (.1)
L (.1)L (.1)
Probab.Probab..0010.0010
.0090.0090
.0090.0090
.7290.7290
.0090.0090
11
11
.0810.0810
.0810.0810
.0810.0810
Example: Evans ElectronicsExample: Evans Electronics
18 18 Slide Slide
Example: Evans ElectronicsExample: Evans Electronics
Using an Excel SpreadsheetUsing an Excel Spreadsheet• Step 1: Select a cell in the worksheet where youStep 1: Select a cell in the worksheet where you
want the binomial probabilities to want the binomial probabilities to appear.appear.
• Step 2: Select the Step 2: Select the InsertInsert pull-down menu. pull-down menu.• Step 3: Choose the Step 3: Choose the FunctionFunction option. option.• Step 4: When the Paste Function dialog box appears:Step 4: When the Paste Function dialog box appears:
Choose Statistical from the Function Category box.
Choose BINOMDIST from the Function Name box. Select OK.
continued
19 19 Slide Slide
Example: Evans ElectronicsExample: Evans Electronics
Using an Excel Spreadsheet (continued)Using an Excel Spreadsheet (continued)• Step 5: When the BINOMDIST dialog box appears:Step 5: When the BINOMDIST dialog box appears:
Enter 1 in the Number_s box (value of x). Enter 3 in the Trials box (value of n). Enter .1 in the Probability_s box (value of p). Enter false in the Cumulative box. [Note: At this point the desired binomial probability of .243 is automatically computed and appears in the right center of the dialog box.] Select OK (and .243 will appear in the worksheet cell requested in Step 1).
20 20 Slide Slide
The Binomial Probability DistributionThe Binomial Probability Distribution
Expected ValueExpected Value
EE((xx) = ) = = = npnp VarianceVariance
Var(Var(xx) = ) = 22 = = npnp(1 - (1 - pp)) Standard DeviationStandard Deviation
Example: Evans ElectronicsExample: Evans Electronics
EE((xx) = ) = = 3(.1) = .3 employees out of 3 = 3(.1) = .3 employees out of 3
Var(Var(xx) = ) = 22 = 3(.1)(.9) = .27 = 3(.1)(.9) = .27
SD( ) ( )x np p 1SD( ) ( )x np p 1
employees 52.)9)(.1(.3)(SD x employees 52.)9)(.1(.3)(SD x
21 21 Slide Slide
MINITAB APPLICATIONMINITAB APPLICATION
Minitab ProgramMinitab Program
22 22 Slide Slide
Poisson Probability DistributionPoisson Probability Distribution
Properties of a Poisson ExperimentProperties of a Poisson Experiment• The probability of an occurrence is the same The probability of an occurrence is the same
for any two intervals of equal length.for any two intervals of equal length.• The occurrence or nonoccurrence in any The occurrence or nonoccurrence in any
interval is independent of the occurrence or interval is independent of the occurrence or nonoccurrence in any other interval.nonoccurrence in any other interval.
23 23 Slide Slide
Poisson Probability DistributionPoisson Probability Distribution
Poisson Probability FunctionPoisson Probability Function
wherewhere
f(x) f(x) = probability of x occurrences in an = probability of x occurrences in an intervalinterval
= mean number of occurrences in an = mean number of occurrences in an intervalinterval
ee = 2.71828 = 2.71828
f xe
x
x( )
!
f x
e
x
x( )
!
24 24 Slide Slide
Example: Mercy HospitalExample: Mercy Hospital
Patients arrive at the emergency room of Patients arrive at the emergency room of MercyMercy
Hospital at the average rate of 6 per hour on Hospital at the average rate of 6 per hour on weekendweekend
evenings. What is the probability of 4 arrivals in evenings. What is the probability of 4 arrivals in 3030
minutes on a weekend evening?minutes on a weekend evening?
Using the Poisson Probability FunctionUsing the Poisson Probability Function
= 6/hour = 3/half-hour, = 6/hour = 3/half-hour, xx = 4 = 4f ( )
( . )!
.43 2 71828
41680
4 3
f ( )( . )
!.4
3 2 718284
16804 3
25 25 Slide Slide
Example: Mercy HospitalExample: Mercy Hospital
Using the Tables of Poisson ProbabilitiesUsing the Tables of Poisson Probabilities
x 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.00 .1225 .1108 .1003 .0907 .0821 .0743 .0672 .0608 .0550 .04981 .2572 .2438 .2306 .2177 .2052 .1931 .1815 .1703 .1596 .14942 .2700 .2681 .2652 .2613 .2565 .2510 .2450 .2384 .2314 .22403 .1890 .1966 .2033 .2090 .2138 .2176 .2205 .2225 .2237 .22404 .0992 .1082 .1169 .1254 .1336 .1414 .1488 .1557 .1622 .16805 .0417 .0476 .0538 .0602 ..0668 .0735 .0804 .0872 .0940 .10086 .0146 .0174 .0206 .0241 .0278 .0319 .0362 .0407 .0455 .05047 .0044 .0055 .0068 .0083 .0099 .0118 .0139 .0163 .0188 .02168 .0011 .0015 .0019 .0025 .0031 .0038 .0047 .0057 .0068 .00819 .0003 .0004 .0005 .0007 .0009 .0011 .0014 .0018 .0022 .002710 .0001 .0001 .0001 .0002 .0002 .0003 .0004 .0005 .0006 .000811 .0000 .0000 .0000 .0000 .0000 .0001 .0001 .0001 .0002 .000212 .0000 .0000 .0000 .0000 .0000 .0000 .0000 .0000 .0000 .0001
x 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.00 .1225 .1108 .1003 .0907 .0821 .0743 .0672 .0608 .0550 .04981 .2572 .2438 .2306 .2177 .2052 .1931 .1815 .1703 .1596 .14942 .2700 .2681 .2652 .2613 .2565 .2510 .2450 .2384 .2314 .22403 .1890 .1966 .2033 .2090 .2138 .2176 .2205 .2225 .2237 .22404 .0992 .1082 .1169 .1254 .1336 .1414 .1488 .1557 .1622 .16805 .0417 .0476 .0538 .0602 ..0668 .0735 .0804 .0872 .0940 .10086 .0146 .0174 .0206 .0241 .0278 .0319 .0362 .0407 .0455 .05047 .0044 .0055 .0068 .0083 .0099 .0118 .0139 .0163 .0188 .02168 .0011 .0015 .0019 .0025 .0031 .0038 .0047 .0057 .0068 .00819 .0003 .0004 .0005 .0007 .0009 .0011 .0014 .0018 .0022 .002710 .0001 .0001 .0001 .0002 .0002 .0003 .0004 .0005 .0006 .000811 .0000 .0000 .0000 .0000 .0000 .0001 .0001 .0001 .0002 .000212 .0000 .0000 .0000 .0000 .0000 .0000 .0000 .0000 .0000 .0001
26 26 Slide Slide
Poisson Probability DistributionPoisson Probability Distribution
The Poisson probability distribution can be The Poisson probability distribution can be used as used as
an approximation of the binomial probabilityan approximation of the binomial probability
distribution when distribution when pp, the probability of success, is , the probability of success, is smallsmall
and and nn, the number of trials, is large., the number of trials, is large.• Approximation is good when Approximation is good when pp << .05 and .05 and nn
>> 20 20• Set Set = = npnp and use the Poisson tables. and use the Poisson tables.
27 27 Slide Slide
MINIAB APPLICATIONMINIAB APPLICATION
Minitab ProgramMinitab Program
28 28 Slide Slide
The End of Discrete Probability DistributionThe End of Discrete Probability Distribution
29 29 Slide Slide
Chapter 3, Part IIChapter 3, Part II Continuous Random Variables Continuous Random Variables
Continuous Random VariablesContinuous Random Variables Normal Probability DistributionNormal Probability Distribution Exponential Probability DistributionExponential Probability Distribution
30 30 Slide Slide
Continuous Probability DistributionsContinuous Probability Distributions
A A continuous random variablecontinuous random variable can assume any can assume any value in an interval on the real line or in a value in an interval on the real line or in a collection of intervals.collection of intervals.
It is not possible to talk about the probability of It is not possible to talk about the probability of the random variable assuming a particular value.the random variable assuming a particular value.
Instead, we talk about the probability of the Instead, we talk about the probability of the random variable assuming a value within a given random variable assuming a value within a given interval.interval.
The probability of the random variable assuming a The probability of the random variable assuming a value within some given interval from value within some given interval from xx11 to to xx22 is is defined to be the defined to be the area under the grapharea under the graph of the of the probability density functionprobability density function betweenbetween x x11 andand x x22..
31 31 Slide Slide
Uniform Probability DistributionUniform Probability Distribution
A random variable is A random variable is uniformly distributeduniformly distributed wheneverwheneverthe probability is proportional to the length of thethe probability is proportional to the length of theinterval. interval. Uniform Probability Density FunctionUniform Probability Density Function
ff((xx) = 1/() = 1/(bb - - aa) for ) for aa << xx << bb = 0 elsewhere= 0 elsewhere
Expected Value of Expected Value of xxE(E(xx) = () = (aa + + bb)/2)/2
Variance of Variance of xx Var(Var(xx) = () = (bb - - aa))22/12/12
where: where: aa = smallest value the variable can = smallest value the variable can assumeassume
bb = largest value the variable can = largest value the variable can assumeassume
32 32 Slide Slide
Example: Slater's BuffetExample: Slater's Buffet
Slater customers are charged for the amount of Slater customers are charged for the amount of saladsalad
they take. Sampling suggests that the amount of they take. Sampling suggests that the amount of saladsalad
taken is uniformly distributed between 5 ounces taken is uniformly distributed between 5 ounces and 15and 15
ounces.ounces. Probability Density Function Probability Density Function
ff((xx) = 1/10 for 5 ) = 1/10 for 5 << xx << 15 15
= 0 elsewhere= 0 elsewhere
wherewhere
xx = salad plate filling weight = salad plate filling weight
33 33 Slide Slide
Example: Slater's BuffetExample: Slater's Buffet
What is the probability that a customer will What is the probability that a customer will taketake
between 12 and 15 ounces of salad?between 12 and 15 ounces of salad?f(x)f(x)
x x55 1010 15151212
1/101/10
Salad Weight (oz.)Salad Weight (oz.)
P(12 < x < 15) = 1/10(3) = .3P(12 < x < 15) = 1/10(3) = .3
34 34 Slide Slide
MINITAB APPLICATIONMINITAB APPLICATION
Minitab ProgramMinitab Program
35 35 Slide Slide
The Normal Probability DistributionThe Normal Probability Distribution
Graph of the Normal Probability Density Graph of the Normal Probability Density FunctionFunction
xx
ff((xx))
36 36 Slide Slide
Normal Probability DistributionNormal Probability Distribution
The Normal CurveThe Normal Curve• The shape of the normal curve is often The shape of the normal curve is often
illustrated as a illustrated as a bell-shaped curvebell-shaped curve. . • The highest point on the normal curve is at The highest point on the normal curve is at
the the meanmean, which is also the , which is also the medianmedian and and modemode of the distribution. of the distribution.
• The normal curve is The normal curve is symmetricsymmetric..• The The standard deviationstandard deviation determines the determines the
width of the curve.width of the curve.• The total The total area under the curvearea under the curve is 1. is 1.• Probabilities for the normal random variable Probabilities for the normal random variable
are given by areas under the curve.are given by areas under the curve.
37 37 Slide Slide
Normal Probability DistributionNormal Probability Distribution
Normal Probability Density FunctionNormal Probability Density Function
wherewhere
= mean= mean
= standard deviation= standard deviation
= 3.14159= 3.14159
ee = 2.71828 = 2.71828
f x e x( ) ( ) / 1
2
2 22
f x e x( ) ( ) / 1
2
2 22
38 38 Slide Slide
Standard Normal Probability DistributionStandard Normal Probability Distribution
A random variable that has a normal A random variable that has a normal distribution with a mean of zero and a distribution with a mean of zero and a standard deviation of one is said to have a standard deviation of one is said to have a standard normal probability distributionstandard normal probability distribution..
The letter The letter z z is commonly used to designate is commonly used to designate this normal random variable.this normal random variable.
Converting to the Standard Normal Converting to the Standard Normal Distribution Distribution
We can think of We can think of zz as a measure of the number as a measure of the number of standard deviations of standard deviations xx is from is from ..
zx
zx
39 39 Slide Slide
Example: Pep ZoneExample: Pep Zone
Pep Zone sells auto parts and supplies including aPep Zone sells auto parts and supplies including a
popular multi-grade motor oil. When the stock of thispopular multi-grade motor oil. When the stock of this
oil drops to 20 gallons, a replenishment order is placed.oil drops to 20 gallons, a replenishment order is placed.
The store manager is concerned that sales are being The store manager is concerned that sales are being
lost due to stockouts while waiting for an order. It haslost due to stockouts while waiting for an order. It has
been determined that leadtime demand is normallybeen determined that leadtime demand is normally
distributed with a mean of 15 gallons and a standarddistributed with a mean of 15 gallons and a standard
deviation of 6 gallons. deviation of 6 gallons.
The manager would like to know the probability of aThe manager would like to know the probability of a
stockout, P(stockout, P(xx > 20). > 20).
40 40 Slide Slide
Standard NormalStandard Normal
DistributionDistribution
zz = ( = (xx - - )/)/
= (20 - 15)/6= (20 - 15)/6
= .83= .83
The Standard Normal table shows an area The Standard Normal table shows an area of .2967 for the region between the of .2967 for the region between the zz = 0 line = 0 line and the and the z z = .83 line above. The shaded tail = .83 line above. The shaded tail area is .5 - .2967 = .2033. The probability of a area is .5 - .2967 = .2033. The probability of a stockout is .2033.stockout is .2033.
00 .83.83
Area = .2967Area = .2967
Area = .5Area = .5
Area = .2033Area = .2033
zz
Example: Pep ZoneExample: Pep Zone
41 41 Slide Slide
Using the Standard Normal Probability TableUsing the Standard Normal Probability Tablez .00 .01 .02 .03 .04 .05 .06 .07 .08 .09
.0 .0000 .0040 .0080 .0120 .0160 .0199 .0239 .0279 .0319 .0359
.1 .0398 .0438 .0478 .0517 .0557 .0596 .0636 .0675 .0714 .0753
.2 .0793 .0832 .0871 .0910 .0948 .0987 .1026 .1064 .1103 .1141
.3 .1179 .1217 .1255 .1293 .1331 .1368 .1406 .1443 .1480 .1517
.4 .1554 .1591 .1628 .1664 .1700 .1736 .1772 .1808 .1844 .1879
.5 .1915 .1950 .1985 .2019 .2054 .2088 .2123 .2157 .2190 .2224
.6 .2257 .2291 .2324 .2357 .2389 .2422 .2454 .2486 .2518 .2549
.7 .2580 .2612 .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
z .00 .01 .02 .03 .04 .05 .06 .07 .08 .09
.0 .0000 .0040 .0080 .0120 .0160 .0199 .0239 .0279 .0319 .0359
.1 .0398 .0438 .0478 .0517 .0557 .0596 .0636 .0675 .0714 .0753
.2 .0793 .0832 .0871 .0910 .0948 .0987 .1026 .1064 .1103 .1141
.3 .1179 .1217 .1255 .1293 .1331 .1368 .1406 .1443 .1480 .1517
.4 .1554 .1591 .1628 .1664 .1700 .1736 .1772 .1808 .1844 .1879
.5 .1915 .1950 .1985 .2019 .2054 .2088 .2123 .2157 .2190 .2224
.6 .2257 .2291 .2324 .2357 .2389 .2422 .2454 .2486 .2518 .2549
.7 .2580 .2612 .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
Example: Pep ZoneExample: Pep Zone
42 42 Slide Slide
Example: Pep ZoneExample: Pep Zone
Using an Excel SpreadsheetUsing an Excel Spreadsheet• Step 1: Select a cell in the worksheet where Step 1: Select a cell in the worksheet where
you you want the normal probability to want the normal probability to appear.appear.
• Step 2: Select the Step 2: Select the InsertInsert pull-down menu. pull-down menu.• Step 3: Choose the Step 3: Choose the FunctionFunction option. option.• Step 4: When the Step 4: When the Paste FunctionPaste Function dialog box dialog box
appears: appears: Choose Choose StatisticalStatistical from the from the Function CategoryFunction Category box. box. Choose Choose NORMDISTNORMDIST from the from the Function NameFunction Name box. box. Select Select OKOK..
continue
43 43 Slide Slide
Example: Pep ZoneExample: Pep Zone
Using an Excel Spreadsheet (continued)Using an Excel Spreadsheet (continued)• Step 5: When the Step 5: When the NORMDISTNORMDIST dialog box dialog box
appears:appears:Enter 20 in the Enter 20 in the xx box. box.
Enter 15 in the Enter 15 in the meanmean box. box.Enter 6 in the Enter 6 in the standard deviationstandard deviation box. box.Enter true in the Enter true in the cumulativecumulative box. box.Select Select OKOK..
At this point, .7967 will appear in the cell selected At this point, .7967 will appear in the cell selected in Step 1, indicating that the probability of demand in Step 1, indicating that the probability of demand during lead time being less than or equal to 20 during lead time being less than or equal to 20 gallons is .7967. The probability that demand will gallons is .7967. The probability that demand will exceed 20 gallons is 1 - .7967 = .2033.exceed 20 gallons is 1 - .7967 = .2033.
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If the manager of Pep Zone wants the probability of aIf the manager of Pep Zone wants the probability of astockout to be no more than .05, what should thestockout to be no more than .05, what should thereorder point be?reorder point be?
Let Let zz.05.05 represent the represent the zz value cutting the tail area value cutting the tail area of .05. of .05.
Area = .05Area = .05
Area = .5 Area = .5 Area = .45 Area = .45
00 zz.05.05
Example: Pep ZoneExample: Pep Zone
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Using the Standard Normal Probability TableUsing the Standard Normal Probability TableWe now look-up the .4500 area in the We now look-up the .4500 area in the Standard Normal Probability table to find the Standard Normal Probability table to find the corresponding corresponding zz.05.05 value. value.
zz.05.05 = 1.645 is a reasonable = 1.645 is a reasonable estimate. estimate.
z .00 .01 .02 .03 .04 .05 .06 .07 .08 .09
.
1.5 .4332 .4345 .4357 .4370 .4382 .4394 .4406 .4418 .4429 .4441
1.6 .4452 .4463 .4474 .4484 .4495 .4505 .4515 .4525 .4535 .4545
1.7 .4554 .4564 .4573 .4582 .4591 .4599 .4608 .4616 .4625 .4633
1.8 .4641 .4649 .4656 .4664 .4671 .4678 .4686 .4693 .4699 .4706
1.9 .4713 .4719 .4726 .4732 .4738 .4744 .4750 .4756 .4761 .4767 .
z .00 .01 .02 .03 .04 .05 .06 .07 .08 .09
.
1.5 .4332 .4345 .4357 .4370 .4382 .4394 .4406 .4418 .4429 .4441
1.6 .4452 .4463 .4474 .4484 .4495 .4505 .4515 .4525 .4535 .4545
1.7 .4554 .4564 .4573 .4582 .4591 .4599 .4608 .4616 .4625 .4633
1.8 .4641 .4649 .4656 .4664 .4671 .4678 .4686 .4693 .4699 .4706
1.9 .4713 .4719 .4726 .4732 .4738 .4744 .4750 .4756 .4761 .4767 .
Example: Pep ZoneExample: Pep Zone
46 46 Slide Slide
The corresponding value of The corresponding value of xx is given by is given by
xx = = + + zz.05.05
= 15 + 1.645(6)= 15 + 1.645(6)
= 24.87= 24.87
A reorder point of 24.87 gallons will place A reorder point of 24.87 gallons will place thethe
probability of a stockout during leadtime at .05. probability of a stockout during leadtime at .05.
Perhaps Pep Zone should set the reorder Perhaps Pep Zone should set the reorder point at 25 gallons to keep the probability point at 25 gallons to keep the probability under .05.under .05.
Example: Pep ZoneExample: Pep Zone
47 47 Slide Slide
Exponential Probability DistributionExponential Probability Distribution
Exponential Probability Density FunctionExponential Probability Density Function
for for xx >> 0, 0, > 0 > 0
wherewhere = mean = mean
ee = 2.71828 = 2.71828 Cumulative Exponential Distribution FunctionCumulative Exponential Distribution Function
wherewhere xx00 = some specific value of = some specific value of xx
f x e x( ) / 1
f x e x( ) / 1
P x x e x( ) / 0 1 o P x x e x( ) / 0 1 o
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The time between arrivals of cars at Al’s The time between arrivals of cars at Al’s Carwash Carwash
follows an exponential probability distribution follows an exponential probability distribution with awith a
mean time between arrivals of 3 minutes. Al mean time between arrivals of 3 minutes. Al would likewould like
to know the probability that the time between to know the probability that the time between twotwo
successive arrivals will be 2 minutes or less.successive arrivals will be 2 minutes or less.
P(P(xx << 2) = 1 - 2.71828 2) = 1 - 2.71828-2/3-2/3 = 1 - .5134 = 1 - .5134 = .4866= .4866
Example: Al’s CarwashExample: Al’s Carwash
49 49 Slide Slide
Example: Al’s CarwashExample: Al’s Carwash
Graph of the Probability Density FunctionGraph of the Probability Density Function
xx
f(x)f(x)
.1.1
.3.3
.4.4
.2.2
1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10
P(x < 2) = area = .4866P(x < 2) = area = .4866
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Relationship Between theRelationship Between thePoisson and Exponential DistributionsPoisson and Exponential Distributions
The continuous exponential probability The continuous exponential probability distribution is related to the discrete Poisson distribution is related to the discrete Poisson distribution.distribution.
The The Poisson distributionPoisson distribution provides an appropriate provides an appropriate description of the description of the number of occurrences per number of occurrences per intervalinterval..
The The exponential distributionexponential distribution provides a provides a description of the description of the length of the interval between length of the interval between occurrencesoccurrences..
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The End of Chapter 3The End of Chapter 3