Chapter 6
Continuous
Probability
Distributions
© 2002 Thomson / South-Western
Slide 6-1
Learning Objectives
• Understand concepts of the uniform
distribution.
• Appreciate the importance of the normal
distribution.
• Recognize normal distribution problems,
and know how to solve them.
• Decide when to use the normal
distribution to approximate binomial
distribution problems, and know how to
work them.
• Decide when to use the exponential
distribution to solve problems in
business, and know how to work them.
© 2002 Thomson / South-Western
Slide 6-2
Uniform Distribution
The uniform distribution is a continuous distribution
in which the same height, of f(X), is obtained over a
range of values.
 1
b  a

f ( x)  
 0


for
a  xb
for
all other values
1
ba
f (x)
Area = 1
a
© 2002 Thomson / South-Western
x
b
Slide 6-3
Example: Uniform Distribution
of Lot Weights
 1
 47  41

f ( x)  
 0


for
for
41  x  47
all other values
1
1

47  41 6
f (x)
Area = 1
41
© 2002 Thomson / South-Western
47
Slide 6-4
x
Example: Uniform Distribution,continued
Mean and Standard Deviation
Mean

a+b
=
2
Mean
41 + 47
88
=

 44
2
2
Standard Deviation
Standard Deviation
ba

12
47  41
6


 1. 732
12
3. 464
© 2002 Thomson / South-Western
Slide 6-5
Example: Uniform Distribution
Probability, continued
x
x
P( x  X  x ) 
ba
2
1
1
2
45  42 1
P( 42  X  45) 

47  41 2
45  42 1

47  41 2
f (x)
Area
= 0.5
41
© 2002 Thomson / South-Western
42
45 47 x
Slide 6-6
The Normal Distribution
• A widely known and much-used
distribution that fits the measurements
of many human characteristics and
most machine-produced items. Many
other variable in business and industry
are normally distributed.
• The normal distribution and its
associated probabilities are an integral
part of statistical quality control
© 2002 Thomson / South-Western
Slide 6-7
Characteristics of the
Normal Distribution
• Continuous distribution
• Symmetrical distribution
• Asymptotic to the
horizontal axis
• Unimodal
• A family of curves
• Total area under the
curve sums to 1.
• Area to right of mean
is 1/2.
• Area to left of mean is 1/2.
© 2002 Thomson / South-Western
1/2
1/2

X
Slide 6-8
Probability Density Function
of the Normal Distribution
1
 2
Where:
f ( x) 
e
1

2
 x 


  
2
  mean of X
  standard deviation of X
 = 3.14159 . . .
e  2.71828 . . .

© 2002 Thomson / South-Western
X
Slide 6-9
Normal Curves for Different
Means and Standard Deviations
5
5
 10
20
30
40
50
© 2002 Thomson / South-Western
60
70
80
90
100
110
120
Slide 6-10
Standardized Normal Distribution
• A normal distribution with
– a mean of zero, and
– a standard deviation of
one
 1
• Z Formula
0
– standardizes any
normal distribution
• Z Score
– computed by the Z
Formula
– the number of standard
deviations which a
value is away from the
mean
© 2002 Thomson / South-Western
Z
X 

Slide 6-11
Z Table
Second Decimal Place in Z
Z 0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.00
0.10
0.20
0.30
0.0000
0.0398
0.0793
0.1179
0.0040
0.0438
0.0832
0.1217
0.0080
0.0478
0.0871
0.1255
0.0120
0.0517
0.0910
0.1293
0.0160
0.0557
0.0948
0.1331
0.0199
0.0596
0.0987
0.1368
0.0239
0.0636
0.1026
0.1406
0.0279
0.0675
0.1064
0.1443
0.0319
0.0714
0.1103
0.1480
0.0359
0.0753
0.1141
0.1517
0.90
1.00
1.10
1.20
0.3159
0.3413
0.3643
0.3849
0.3186
0.3438
0.3665
0.3869
0.3212
0.3461
0.3686
0.3888
0.3238
0.3485
0.3708
0.3907
0.3264
0.3508
0.3729
0.3925
0.3289
0.3531
0.3749
0.3944
0.3315
0.3554
0.3770
0.3962
0.3340
0.3577
0.3790
0.3980
0.3365
0.3599
0.3810
0.3997
0.3389
0.3621
0.3830
0.4015
2.00
0.4772
0.4778
0.4783
0.4788
0.4793
0.4798
0.4803
0.4808
0.4812
0.4817
3.00
3.40
3.50
0.4987
0.4997
0.4998
0.4987
0.4997
0.4998
0.4987
0.4997
0.4998
0.4988
0.4997
0.4998
0.4988
0.4997
0.4998
0.4989
0.4997
0.4998
0.4989
0.4997
0.4998
0.4989
0.4997
0.4998
0.4990
0.4997
0.4998
0.4990
0.4998
0.4998
© 2002 Thomson / South-Western
Slide 6-12
Table Lookup of a
Standard Normal Probability
P(0  Z  1)  0. 3413
Z
-3
-2
-1
0
1
© 2002 Thomson / South-Western
2
3
0.00
0.01
0.02
0.00
0.10
0.20
0.0000 0.0040 0.0080
0.0398 0.0438 0.0478
0.0793 0.0832 0.0871
1.00
0.3413 0.3438 0.3461
1.10
1.20
0.3643 0.3665 0.3686
0.3849 0.3869 0.3888
Slide 6-13
Applying the Z Formula:
Example, Assume….
 = 485, and  = 105
P( 485  X  600 )  P( 0  Z  1.10 ) . 3643
X is normally distributed with
For X = 485,
Z=
X -  485  485

0

105
Z
0.00
0.01
0.02
0.00
0.10
0.0000 0.0040 0.0080
0.0398 0.0438 0.0478
For X = 600,
1.00
0.3413 0.3438 0.3461
X -  600  485
Z=

 1.10

105
1.10
0.3643 0.3665 0.3686
1.20
0.3849 0.3869 0.3888
© 2002 Thomson / South-Western
Slide 6-14
Normal Approximation
of the Binomial Distribution
• The normal distribution can be used to
approximate binomial probabilities
• Procedure
– Convert binomial parameters to normal
parameters
– Does the interval m3s lie between 0 and
n? If so, continue; otherwise, do not use
the normal approximation.
– Correct for continuity
– Solve the normal distribution problem
© 2002 Thomson / South-Western
Slide 6-15
Using the Normal Distribution to
Work Binomial Distribution Problems
• The normal distribution can be used to
approximate the probabilities in binomial
distribution problems that involve large
values of n.
• To work a binomial problem by the
normal distribution requires conversion
of the n and p of the binomial
distribution to the µ and s of the normal
distribution.
© 2002 Thomson / South-Western
Slide 6-16
Normal Approximation of Binomial:
Parameter Conversion
• Conversion equations
  n p
  n pq
• Conversion example:
Given that X has a binomial distribution
, find
P( X  25| n  60 and p . 30 ).
  n  p  (60 )(. 30 )  18
  n  p  q  (60 )(. 30 )(. 70 )  3. 55
© 2002 Thomson / South-Western
Slide 6-17
Normal Approximation of Binomial:
Interval Check
  3  18  3(355
. )  18  10.65
  3  7.35
  3  28.65
0
10
20
© 2002 Thomson / South-Western
30
40
50
60
n
70
Slide 6-18
Normal Approximation of Binomial:
Correcting for Continuity
Values
Being
Determined
Correction
X
X
X
X
X
X
+.50
-.50
-.50
+.05
-.50 and +.50
+.50 and -.50
© 2002 Thomson / South-Western
The binomial probability ,
P( X  25| n  60 and p . 30 )
is approximated by the normal probability
P(X  24.5|   18 and   3. 55).
Slide 6-19
Normal Approximation of Binomial:
Graphs
0.12
0.10
0.08
0.06
0.04
0.02
0
6
8
10 12 14 16 18 20 22 24 26 28 30
© 2002 Thomson / South-Western
Slide 6-20
Normal Approximation of Binomial:
Computations
X
P(X)
25
26
27
28
29
30
31
32
33
Total
0.0167
0.0096
0.0052
0.0026
0.0012
0.0005
0.0002
0.0001
0.0000
0.0361
© 2002 Thomson / South-Western
The normal approximation,
P(X  24.5|   18 and   355
. )
24.5  18 

 P Z 



355
.
 P( Z  183
. )
.5  P 0  Z  183
. 
.5.4664
.0336
Slide 6-21
Exponential Distribution
•
•
•
•
•
•
•
Continuous
Family of distributions
Skewed to the right
X varies from 0 to infinity
Apex is always at X = 0
Steadily decreases as X gets larger
Probability function
 X
f (X)  e
© 2002 Thomson / South-Western
for X  0,   0
Slide 6-22
Graphs of Selected Exponential
Distributions
2.0




1.8
1.6
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
0
1
2
© 2002 Thomson / South-Western
3
4
5
6
7
8
Slide 6-23
Exponential Distribution Example:
Probability Computation
1.2

1.0
0.8
X 0
P X  X 0   e
(12
. )(2)
P X  2|   12
. e
.0907
0.6
0.4
0.2
0.0
0
1
© 2002 Thomson / South-Western
2
3
4
5
Slide 6-24