Transcript Continuous Probability Distributions

Continuous Probability
Distributions
Introduction to Business Statistics, 5e
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Probability for a
Continuous Random Variable
Introduction to Business Statistics, 5e
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(c)2000 South-Western College
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Figure 6.1
Properties of a
Normal Distribution
• Continuous Random Variable
• Symmetrical in shape (Bell shaped)
• The probability of any given range of
numbers is represented by the area under
the curve for that range.
• Probabilities for all normal distributions are
determined using the Standard Normal
Distribution.
Introduction to
Business Statistics, 5e
Kvanli/Guynes/Pavur
(c)2000 South-Western
College Publishing
Probability Density Function for
Normal Distribution
x


1
1 (
)
2
f (x) 
e

 2
Introduction to Business Statistics, 5e
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(c)2000 South-Western College
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2
Figure 6.2
Introduction to Business Statistics, 5e
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Figure 6.3
Introduction to Business Statistics, 5e
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Figure 6.4
Introduction to Business Statistics, 5e
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Figure 6.5
Introduction to Business Statistics, 5e
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Figure 6.6
Introduction to Business Statistics, 5e
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Determining the Probability for a
Standard Normal Random Variable
• Figures 6.10-6.13
• P(- Z  1.62) = .5 + .4474 = .9474
• P(Z > 1.62) = 1 - P(- Z  1.62) =
1 - .9474 = .0526
Introduction to Business Statistics, 5e
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(c)2000 South-Western College
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Figure 6.10
Introduction to Business Statistics, 5e
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Figure 6.11
Introduction to Business Statistics, 5e
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Determining the probability of
any Normal Random Variable
Fig 6.20
Introduction to Business Statistics, 5e
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Interpreting Z
• Example 6.2 Z = - 0.8 means that the value
360 is .8 standard deviations below the mean.
• A positive value of Z designates how may
standard deviations () X is to the right of the
mean ().
• A negative value of Z designates how may
standard deviations () X is to the left of the
mean ().
Introduction to Business Statistics, 5e
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Example 6.5
Referring to Example 6.2, after how many hours will
80% of the Evergol bulbs burn out?
P(Z < .84) =
.5 + .2995 =
.7995  .8
Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur
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Figure 6.26
Figure 6.26
Introduction to Business Statistics, 5e
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x o  400
Z
 .84
50
x o  400  50(.84)  42
x o  400  42  442
Continuous Uniform Distribution
• The probability of a given range of values is
proportional to the width of the range.
• Distribution Mean:
ab
 
2
b– a
• Standard Deviation:  
12
Introduction to Business Statistics, 5e
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Figure 6.35
Introduction to Business Statistics, 5e
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Figure 6.36
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Exponential Distribution
Applications:
• Time between arrivals to a queue (e.g. time
between people arriving at a line to check
out in a department store. (People, machines, or
telephone calls may wait in a queue)
• Lifetime of components in a machine
Introduction to Business Statistics, 5e
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Mean and Standard Deviation
P(X  x0 ) 1 – e–Ax 0
where A 1/  ,
Mean:
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and
1
=
A
Standard Deviation:
Introduction to Business Statistics, 5e
for x0  0
1
 .
A
Figure 6.39
P (X  x0 )  1 – e– Ax0
Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur
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for x0  0
1
1
where A  1 / ,  = , and   .
A
A