06 Probability Distribution

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Slide 2003 Thomson/South-Western

Chapter 6Continuous Probability Distributions

Uniform Probability Distribution

Normal Probability Distribution

Exponential Probability Distribution

x

f(x)

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Continuous Probability Distributions

A continuous random variable can assume any valuein an interval on the real line or in a collection ofintervals.

It is not possible to talk about the probability of therandom variable assuming a particular value.

Instead, we talk about the probability of the randomvariable assuming a value within a given interval.

The probability of the random variable assuming avalue within some given interval from x1 to x2 is

defined to be the area under the graph of theprobability density function between x1and x2.



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The normal probability distribution is the most

important distribution for describing a continuousrandom variable.

It has been used in a wide variety of applications:

Heights and weights of people

Test scores

Scientific measurements

Amounts of rainfall

It is widely used in statistical inference



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Normal Probability Density Function

where:

= population mean

= population standard deviation

= 3.14159

e = 2.71828

f x e x( ) ( ) / 1

2

22

2



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Graph of the Normal Probability Density Function

x

f(x)



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Characteristics of the Normal ProbabilityDistribution

The distribution is symmetric, and is oftenillustrated as a bell-shaped curve.

Two parameters, (mean) and (standarddeviation), determine the location and shape ofthe distribution.

The highest point on the normal curve is at themean, which is also the median and mode.

The mean can be any numerical value: negative,zero, or positive.

continued



7/197Slide 2003 Thomson/South-Western


Characteristics of the Normal Probability

Distribution

The standard deviation determines the width ofthe curve: larger values result in wider, flattercurves.

= 10

= 50





Characteristics of the Normal Probability Distribution

The total area under the curve is 1 (.5 to the left ofthe mean and .5 to the right).

Probabilities for the normal random variable aregiven by areas under the curve.





Characteristics of the Normal Probability

Distribution

68.26% of values of a normal random variable arewithin +/- 1standard deviation of its mean.

95.44% of values of a normal random variable are

within +/- 2standard deviations of its mean.

99.72% of values of a normal random variable arewithin +/- 3standard deviations of its mean.




A random variable that has a normal distribution

with a mean of zero and a standard deviation of oneis said to have a standard normal probabilitydistribution.

The letter z is commonly used to designate this

normal random variable. Converting to the Standard Normal Distribution

We can think of z as a measure of the number ofstandard deviations x is from .

Standard Normal Probability Distribution

zx




Example: Pep Zone

Standard Normal Probability Distribution

Pep Zone sells auto parts and supplies including a

popular multi-grade motor oil. When the stock of this

oil drops to 20 gallons, a replenishment order is placed.

The store manager is concerned that sales are beinglost due to stock outs while waiting for an order. It

has been determined that lead time demand isnormally distributed with a mean of 15 gallons and astandard deviation of 6 gallons.

The manager would like to know the probability of a

stockout, P(x > 20).




Step 1

Convert x into standard z-value

z = (x - )/

= (20 - 15)/6

= .83

Example: Pep Zone

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Using the Standard Normal Probability Table

Example: Pep Zone

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



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Step 2

Find area under curve from z = 0 to z = 0.83

Example: Pep Zone

0 .83

Area = .2967

Area = .5z



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Step 3 : Find Probability of Stockout (shortage)

The Standard Normal table shows an area of .2967 forthe region between the z = 0 and z = .83 lines below.The shaded tail area is .5 - .2967 = .2033. Theprobability of a stock-out is .2033.

P ( x > 20 ) = .2033

Example: Pep Zone

0 .83

Area = .2967

Area = .5

Area = .5 - .2967= .2033

z



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FINDING REORDER POINT (x) FOR SPECIFIED SERVICE LEVEL OF 95%

SHORATGES OCCURRENCE 5%

If the manager of Pep Zone wants the probability of a stockout to be no morethan .05, what should the reorder point be?

Let z.05 represent the z value cutting the .05 tail area.

What is the value of

zfor which Area under curveis 0.45

Area = .5 Area = .45

0 z.05


0.05 xz

Example: Pep Zone (Inverse z-transform)


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How to find value of z for specific AREA UNDERCURVE

We now look-up the .4500 area in the StandardNormal Probability table to find the correspondingz.05 value.

z.05 = 1.645 is a reasonable 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

. . . . . . . . . . .




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How to find re-order point ?

Let z.05 represent the z value cutting the .05 tail area.

What is the value of

zfor which Area under curveis 0.45, answer = 1.645

Area = .5 Area = .45

0


z.05

0.05

151.645

6

xz



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Finding re-order point

The corresponding value of x is given by

x = + z.05= 15 + 1.645(6)

= 24.87

A reorder point of 24.87 gallons will place the

probability of a stock-out during lead time at .05Perhaps Pep Zone should set the reorder point at

25 gallons to keep the probability under .05

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06 Probability Distribution