1 Chapter 6 Continuous Probability Distributions Uniform Probability Distribution Normal Probability...

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Chapter 6 Continuous Probability Distributions

Uniform Probability DistributionNormal Probability DistributionExponential Probability Distribution

xx

ff((xx))

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Continuous Probability Distributions

A continuous random variable can assume any value in an interval on the real line or in a collection of intervals.It is not possible to talk about the probability of the random variable assuming a particular value.Instead, we talk about the probability of the random variable assuming a value within a given interval.The probability of the random variable assuming a value within some given interval from x1 to x2 is defined to be the area under the graph of the probability density function between x1 and x2.

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A random variable is uniformly distributed whenever the probability is proportional to the interval’s length. Uniform Probability Density Function

f(x) = 1/(b - a) for a < x < b

= 0 elsewhere

where: a = smallest value the variable can assume

b = largest value the variable can assume

Uniform Probability Distribution

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Expected Value of x

E(x) = (a + b)/2

Variance of x

Var(x) = (b - a)2/12 where: a = smallest value the variable can assume

b = largest value the variable can assume

Uniform Probability Distribution

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Example: Slater's Buffet

Uniform Probability DistributionSlater customers are charged for the

amount of salad they take. Sampling suggests that the amount of salad taken is uniformly distributed between 5 ounces and 15 ounces.

The probability density function is f(x) = 1/10 for ______ < x <

_______ = 0 elsewhere

where:x = salad plate filling weight

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Uniform Probability Distributionfor Salad Plate Filling Weight

f(x)f(x)

x x55 1010 1515

1/101/10

Salad Weight (oz.)Salad Weight (oz.)

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Uniform Probability DistributionWhat is the probability that a customer

will take 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) = ____P(12 < x < 15) = 1/10(3) = ____

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Expected Value of xE(x) = (a + b)/2 = (5 + 15)/2 = ______

Variance of x Var(x) = (b - a)2/12

= (15 – 5)2/12 = ______

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Normal Probability Distribution

The normal probability distribution is the most important distribution for describing a continuous random variable.It has been used in a wide variety of applications:– Heights and weights of people– __________– Scientific measurements– Amounts of rainfall

It is widely used in statistical inference

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Normal Probability Density Function

where: = mean = standard deviation = 3.14159 e = 2.71828

f x e x( ) ( ) / 12

2 2 2

f x e x( ) ( ) / 1

2

2 2 2

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Graph of the Normal Probability Density Function

xx

ff((xx))

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Characteristics of the Normal Probability Distribution– The distribution is symmetric, and is often

illustrated as a bell-shaped curve. – Two parameters, (mean) and (standard

deviation), determine the location and shape of the distribution.

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

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

… continued

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Characteristics of the Normal Probability Distribution– The standard deviation determines the

width of the curve: larger values result in wider, flatter curves.

= 10= 10

= 50= 50

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Characteristics of the Normal Probability Distribution– The total area under the curve is 1 (.5 to the

left of the mean and .5 to the right).– Probabilities for the normal random variable

are given by areas under the curve.

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Characteristics of the Normal Probability Distribution– 68.26% of values of a normal random

variable are within +/- 1 standard deviation of its mean.

– 95.44% of values of a normal random variable are within +/- 2 standard deviations of its mean.

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

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A random variable that has a normal distribution with a mean of zero and a standard deviation of one is said to have a standard normal probability distribution.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 of standard deviations x is from .

Standard Normal Probability Distribution

zx

zx

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Example: Pep Zone

Standard Normal Probability DistributionPep 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 being lost due to stockouts while waiting for an order (leadtime). It has been determined that leadtime demand is normally distributed with a mean of 15 gallons and a standard deviation of 6 gallons.

The manager would like to know the probability of a

stockout, P(x > 20).

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Standard Normal Probability DistributionThe Standard Normal table shows an area of ____ for the region between the z = 0 and z = ___ lines below. The shaded tail area is .5 - .2967 = .2033. The probability of a stock-

out is _____. z = (x - )/ = (20 - 15)/6

= .83

Example: Pep Zone

00 .83.83

Area = .2967Area = .2967

Area = .5Area = .5

Area = .5 - .2967Area = .5 - .2967 = .2033= .2033

zz

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Using the Standard Normal Probability Table (e.g., Appendix B, Table 1)

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

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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Standard Normal Probability DistributionIf the manager of Pep Zone wants the probability of a stockout to be no more than .05, what should the reorder point be?

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

area.

Example: Pep Zone

Area = .05Area = .05

Area = .5 Area = .5 Area = .45 Area = .45

00 zz.05.05

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Using the Standard Normal Probability TableWe now look-up the .4500 area in the Standard Normal Probability table to find the corresponding z.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

. . . . . . . . . . .

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 Zone

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Standard Normal Probability DistributionThe corresponding value of x is given by

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

= _______A reorder point of ______ gallons will

place the probability of a stockout during leadtime at .05. Perhaps Pep Zone should set the reorder point at 25 gallons to keep the probability under .05.

Example: Pep Zone

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Exponential Probability Distribution

The exponential probability distribution is useful in describing the time it takes to complete a task.The exponential random variables can be used to describe:– Time between vehicle arrivals at a toll booth– Time required to complete a questionnaire– Distance between major defects in a

highway

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Exponential Probability Density Function

for x > 0, > 0

where: = mean e = 2.71828


f x e x( ) / 1

f x e x( ) / 1

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Cumulative Exponential Distribution Function

where: x0 = some specific value of x


P x x e x( ) / 0 1 o P x x e x( ) / 0 1 o

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Exponential Probability DistributionThe time between arrivals of cars at Al’s

Carwash follows an exponential probability distribution with a mean time between arrivals of 3 minutes. Al would like to know the probability that the time between two successive arrivals will be 2 minutes or less.

P(x < 2) = 1 - 2.71828-2/3 = 1 - .5134 = _____

Example: Al’s Carwash

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Example: Al’s Carwash

Graph 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

Time Between Successive Arrivals (mins.)Time Between Successive Arrivals (mins.)

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Relationship between the Poissonand Exponential Distributions

(If) the Poisson distribution(If) the Poisson distributionprovides an appropriate descriptionprovides an appropriate description

of the number of occurrencesof the number of occurrencesper intervalper interval

(If) the Poisson distribution(If) the Poisson distributionprovides an appropriate descriptionprovides an appropriate description

of the number of occurrencesof the number of occurrencesper intervalper interval

(If) the exponential distribution(If) the exponential distributionprovides an appropriate descriptionprovides an appropriate description

of the length of the intervalof the length of the intervalbetween occurrencesbetween occurrences

(If) the exponential distribution(If) the exponential distributionprovides an appropriate descriptionprovides an appropriate description

of the length of the intervalof the length of the intervalbetween occurrencesbetween occurrences

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End of Chapter 6

1 Chapter 6 Continuous Probability Distributions Uniform Probability Distribution Normal Probability...

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