Chapter 7 Outline
Introduction
The previous chapter dealt with discrete probability distributions. Recall for a discrete
distribution the outcome can assume only a specific set of values. For example, the number of
correct responses to ten true-false questions can only be the numbers 0, 1, 2, ...., 10.
This chapter continues our study of probability distributions by examining the continuous
probability distribution. Recall that a continuous probability distribution can assume an infinite
number of values within a given range. As an example, the weights for a sample of small engine
blocks are: 54.3, 52.7, 53.1 and 53.9 pounds.
We consider two families of continuous probability distributions, the uniform probability
distribution and the normal probability distribution. Both of these distributions describe the
likelihood of a continuous random variable that has an infinite number of possible values within a
specified range.
An example of a uniform probability distribution is the flight time between Detroit and Chicago.
Suppose the time to fly from Detroit to Chicago is uniformly distributed within a range of 55
minutes to 75 minutes. We can determine the probability that we can fly from Detroit to Chicago
in less than 60 minutes. Flight time is measured on a continuous scale.
The normal probability distribution is described by its mean and standard deviation. Suppose the
life of an automobile battery follows the normal probability distribution with a mean of 36
months and a standard deviation of 3 months. We can determine the probability that a battery will
last between 36 and forty months. Life of a battery is measured on a continuous scale.
Characteristics of the Uniform Probability Distribution
The uniform probability distribution is the simplest distribution for a continuous random variable.
Uniform probability distribution: A continuous probability distribution with its
values spread evenly over a range of values that are rectangular in shape and are
defined by minimum and maximum values.
A uniform distribution is shown in Chart 7-1. The distribution’s shape is rectangular and has a
minimum value of “a ” and a maximum value of “b”. The height of the distribution is uniform for
all values between “a ” and “b”. This implies that all the values in the range are equally likely.
P(X)

ba
X
a
b
The mean of a uniform distribution is located in the middle of the interval between the minimum
value of “a ” and a maximum value of “b”. It is calculated using Formula [7–1]:
Mean of a Uniform Distribution
=
ab
2
7  1
For example: Suppose that the time to fly from Detroit to Chicago is uniformly distributed within
a range of 55 minutes minimum to 75 minutes maximum. The mean is found by using Formula
[7–1]:

a  b 55  75 130


 65
2
2
2
Thus the mean flight time is 65 minutes.
The standard deviation describes the dispersion of a distribution. In a uniform distribution, the
standard deviation is also related to the interval between the minimum value of “a ” and a
maximum value of “b”. It is calculated using Formula [7–2]:
Standard Deviation of a Uniform Distribution
=
(b  a ) 2
12
 7  2
For the flight time from Detroit to Chicago example the standard deviation is calculated using
Formula
[7–2]
=
(b  a) 2
(75  55) 2
(20) 2
400



 33.3333  5.77  5.8
12
12
12
12
Thus the standard deviation for the flight is 5.8 minutes.
Another key element of the uniform distribution is the height: P(x). The height is the same for all
values of the random variable “x “.The equation for the height of a uniform probability
distribution is given in Formula [7–3]:
Uniform Distribution
P( x) =
1
if a  x  b and 0 elsewhere
(b  a)
7  3
In Chapter 6, we discussed the fact that probability distributions are useful when making
probability statements concerning the values of a random variable. Also for continuous random
variables, areas within the distribution represent probabilities. Recall that:
 P( x)  1 and 0  P( x)  1 for all values of x.
The relationship between area and probabilities is applied to the uniform distribution and its
rectangular shape using the area of a rectangle formula. Recall that:
Area of a rectangle = Height  Base
For a uniform distribution the height is P( x) =
1
and the length is (b  a) . If we calculate
(b  a)
the area of the rectangle we have:
Area of a rectangle = Base  Height
=
1
 (b  a)  1
(b  a)
Thus for any uniform distribution, the area under the curve is always 1.
For the flight time from Detroit to Chicago example the area is:
Area of a rectangle = Height  Base
=
1
1
20
 (b  a) 
 (75  55) 
1
(b - a)
(75  55)
20
The Family of Normal Probability Distributions
The Greek letter  (lower case), represents the mean of a normal probability distribution and the
Greek letter  represents the standard deviation.
Normal probability distribution: A continuous probability distribution uniquely
determined by μ and .
The major characteristics of the normal probability distribution are:
1. The normal probability distribution is “bell-shaped” and the mean, median, and mode are
all equal and are located in the center of the distribution. Exactly one-half of the area under
the normal curve is above the center and one-half of the area is below the center.
2. The distribution is symmetrical about the mean. A vertical line drawn at the mean divides
the distribution into two equal halves and these halves possess exactly the same shape.
3. It is asymptotic. That is, the tails of the curve approach the X-axis but never actually touch
it.
4. A normal probability distribution is completely described by its mean and standard
deviation. This indicates that if the mean and standard deviation are known, a normal
probability distribution can
be constructed and its curve
drawn.
5. There is a “family” of
normal probability
distributions. This means
there is a different normal
probability distribution for
each combination of  and .
These characteristics are summarized in the graph.
The Standard Normal Probability Distribution
As noted in the previous discussion, there are many normal probability distributions — a different
one for each pair of values for a mean and a standard deviation. This principle makes the normal
probability distribution applicable to a wide range of real-world situations. However, since there
are an infinite number of probability distributions, it would be awkward to construct tables of
probabilities for so many different normal probability distributions. An efficient method for
overcoming this difficulty is to standardize each normal probability distribution.
Standard normal probability distribution: A normal probability distribution with
a mean of 0 and a standard deviation of 1.
An actual distribution is converted to a standard normal probability distribution using a z value.
z value: The signed distance between a selected value designated X, and the
population mean,  , divided by the population standard deviation, .
The formula for a specific standardized z value is text formula [7–5]:
z=
Standard Normal Value
X -
7  5

Where:
X is the value of any particular observation or measurement.
 is the mean of the distribution.
 is the standard deviation of the distribution.
z is the standardized normal value, usually called the z value.
Applications of the Standard Normal Probability Distribution
To obtain the probability of a value falling in the interval between the variable of interest (X) and
the mean (), we first compute the distance between the value (X) and the mean (). Then we
express that difference in units of the standard deviation by dividing (X  ) by the standard
deviation. This process is called standardizing.
To illustrate the probability of a value being between a selected X value and the mean , suppose
the mean useful life of a car battery is 36 months, with a standard deviation of 3 months. What is
the probability that such a battery will last between 36 and 40 months?
The first step is to convert the 40 months to an equivalent standard normal value, using formula
X -  40  36 4
[7–5]. The computation is: z =

  1.33

3
3
z
0.00
0.01
0.02
0.03
0.04
!
!
!
!
!
!
!
!
0.05
Next refer to Appendix B.1, a table
for the areas under the normal curve.
A part of the table in Appendix B.1 is
shown at the right.
1.0
1.1
1.2
1.3
1.4
!
!
!
!
0.3665
0.3869
0.4049
0.4207
0.3686
0.3888
0.4066
0.4222
0.3708
0.3907
0.4082
0.4236
0.3729
0.3925
0.4099
0.4251
To use the table, the z value of 1.33 is
split into two parts, 1.3 and 0.03. To obtain the probability go down the left-hand column to 1.3,
then move over to the column headed 0.03 and read the probability. It is 0.4082.
The probability that a battery will last between 36 and 40 months is 0.4082. Other probabilities
may be calculated, such as more than 46 months, and less than 33 months. Further details are
given in Problems 1 through 5.
Empirical Rule
Before examining various applications
of the standard normal probability
distribution, three areas under the
normal curve will be considered
which will be used in the following
chapters. They were also called the
Empirical Rule in Chapter 3.
1. About 68 percent of the area
under the normal curve is
within plus one and minus one
standard deviation of the mean.
This can be written as   1 .
2. About 95 percent of the area under the normal curve is within plus and minus two standard
deviations of the mean, written   2
3. Practically all of the area under the normal curve is within three standard deviations of the
mean, written   3 .
The estimates given above are the same as those shown on the diagram.