Transcript Document

Chapter 6
Continuous Probability Distributions
Uniform Probability Distribution
Normal Probability Distribution
Exponential Probability Distribution
f(x)

x
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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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Uniform Probability Distribution
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
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Uniform Probability Distribution
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
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Example: Slater's Buffet
Uniform Probability Distribution
Slater 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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Example: Slater's Buffet
Uniform Probability Distribution
for Salad Plate Filling Weight
f(x)
1/10
x
5
10
15
Salad Weight (oz.)
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Example: Slater's Buffet
Uniform Probability Distribution
What is the probability that a customer will
take between 12 and 15 ounces of salad?
f(x)
P(12 < x < 15) = 1/10(3) = ____
1/10
x
5
10 12
15
Salad Weight (oz.)
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Example: Slater's Buffet
Expected Value of x
E(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 Distribution
Normal Probability Density Function
1
 ( x   ) 2 / 2 2
f ( x) 
e
2 
where:
 = mean
 = standard deviation
 = 3.14159
e = 2.71828
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Normal Probability Distribution
Graph of the Normal Probability Density Function
f(x)

x
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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
 = 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.
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Standard Normal Probability Distribution
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
z
x

We can think of z as a measure of the number of
standard deviations x is from .
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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 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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Example: Pep Zone
Standard Normal Probability Distribution
The 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 stockout is _____.
Area = .2967
z = (x - )/
= (20 - 15)/6
= .83
Area = .5 - .2967
= .2033
Area = .5
0
.83
z
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Example: Pep Zone
Using the Standard Normal Probability Table (e.g.,
Appendix B, Table 1)
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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Example: Pep Zone
Standard Normal Probability Distribution
If the manager of Pep Zone wants the probability of a
stockout to be no more
than .05, what should the
reorder point be?
Area = .05
Area = .5 Area = .45
z.05
0
Let z.05 represent the z value cutting the .05 tail area.
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Example: Pep Zone
Using the Standard Normal Probability Table
We now look-up the .4500 area in the Standard
Normal Probability table to find the corresponding
z.05 value.
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.05 = 1.645 is a reasonable estimate.
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Example: Pep Zone
Standard Normal Probability Distribution
The 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.
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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 Distribution
Exponential Probability Density Function
f ( x) 
1

e  x /  for x > 0,  > 0
where:
 = mean
e = 2.71828
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Exponential Probability Distribution
Cumulative Exponential Distribution Function
P ( x  x0 )  1  e  xo / 
where:
x0 = some specific value of x
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Example: Al’s Carwash
Exponential Probability Distribution
The 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 = _____
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Example: Al’s Carwash
Graph of the Probability Density Function
f(x)
.4
.3
P(x < 2) = area = .4866
.2
.1
x
1
2
3
4
5
6
7
8
9 10
Time Between Successive Arrivals (mins.)
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Relationship between the Poisson
and Exponential Distributions
(If) the Poisson distribution
provides an appropriate description
of the number of occurrences
per interval
(If) the exponential distribution
provides an appropriate description
of the length of the interval
between occurrences
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End of Chapter 6
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