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Chapter 6 Continuous Probability Distributions
Uniform Probability DistributionNormal Probability DistributionStandard Normal Probability DistributionExponential Probability Distribution
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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.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
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 5 < x < 15 = 0 elsewhere
where:x = salad plate filling weight
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Example: Slater's Buffet
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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Example: Slater's Buffet
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) = .3P(12 < x < 15) = (1/10)(3) = .3
Area as a measure of Probability
Area = Width * Height In previous example the area will be
Area = 3 (1/10) = 0.3
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Example: Slater's Buffet
Expected Value of xE(x) = (a + b)/2 = (5 + 15)/2 = 10
Variance of x Var(x) = (b - a)2/12
= (15 – 5)2/12 = 8.33
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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– Test scores– Amounts of rainfall
It is widely used in statistical inference
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Normal Probability Distribution
Normal Probability Density Function
where: = mean = standard deviation = 3.14159 e = 2.71828
22 2/)(
2
1)(
xexf
22 2/)(
2
1)(
xexf
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Normal Probability Distribution
Graph of the Normal Probability Density Function
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ff((xx))
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Normal Probability Distribution
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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Normal Probability Distribution
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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Normal Probability Distribution
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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Normal Probability Distribution
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.
Normal Probability Distribution
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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
Standard Normal Probability Distribution
zx
zx
Standard Normal Probability Distribution
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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 refill order is placed.
The store manager is concerned that sales are being lost due to stock outs while waiting for an order (lead time). It has been determined that lead time 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
Stock out, P(x > 20).
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Standard Normal Probability DistributionThe Standard Normal table shows an area of .2967 for the region between the z = 0 and z = .83 lines below. The shaded tail area is .5 - .2967 = .2033. The probability of a stock-
out is .2033. 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)
= 24.87A reorder point of 24.87 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
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Exponential Probability Density Function
for x > 0, > 0
where: = mean e = 2.71828
Exponential Probability Distribution
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
Exponential Probability Distribution
/0
o1)( xexxP /0
o1)( xexxP
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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 = .4866
Example: Al’s Carwash
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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