Continuous Probability Distributions

Chapter 7

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Learning Objectives
LO7-1 Describe the uniform probability
distribution and use it to calculate
probabilities
LO7-2 Describe the characteristics of a normal
probability distribution
LO7-3 Describe the standard normal probability
distribution and use it to calculate
probabilities
LO7-4 Describe the exponential probability
distribution and use it to calculate
probabilities

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Uniform Distribution
 The uniform distribution characteristics
 It is rectangular in shape
 The mean and the median are equal
 It is completely described by its minimum value a and
its maximum value b

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Uniform Distribution Formulas
 The mean and standard deviation of a uniform
distribution are computed as follows

 The following equation describes the region from a to b

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Uniform Distribution Example
Southwest Arizona State University provides bus service to
students while they are on campus.A bus arrives at the North
Main Street and College Drive stop every 30 minutes between 6
a.m. and 11 p.m. during weekdays. Students arrive at the bus stop
at random times.The time that a student waits is uniformly
distributed from 0 to 30 minutes.

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Uniform Distribution Example (2 of 4)
The area of the uniform distribution is found by multiplying height × base
Area = (30-0) = 1.00
The mean is = = 15
The standard deviation is = = 8.66

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Uniform Distribution Example (3 of 4)

To find the probability that a student will wait more than 25

minutes, find the area between 25 and 30 minutes.
P(25 < wait time < 30) = (height)(base) = (5) = .1667

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Uniform Distribution Example (4 of 4)

To find the probability that a student will wait between 10 and

20 minutes, find the area between 10 and 20 minutes.
P(10 < wait time < 20) = (height)(base) = (10) = .3333

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Normal Probability Distribution
 The normal probability distribution is a continuous distribution
with the following characteristics:
 It is bell-shaped and has a single peak at the center of the
distribution
 The distribution is symmetric
 It is asymptotic, meaning the curve approaches but never
touches the X-axis
 It is completely described by its mean and standard
deviation
 There is a family of normal probability distributions
 Another normal probability distribution is created when
either the mean or the standard deviation changes
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The Normal Curve

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Family of Normal Probability Distributions

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Standard Normal Probability Distribution
 The standard normal probability distribution is a
particular normal distribution
 It has a mean of 0 and a standard deviation of 1

z VALUE The signed distance between a selected value, designated x,

and the mean, μ, divided by the standard deviation, σ.

 Any normal probability distribution can be converted to

the standard normal probability distribution with the
following formula:

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Areas Under the Normal Curve
 Here is a portion of the “z” Table

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Standard Normal Probability Example
Rideshare services are available internationally. A customer uses a smartphone
app to request a ride. Then, a driver receives the request, picks up the
customer, and takes the customer to the desired location. No cash is involved;
the payment for the transaction is handled digitally.

Suppose the weekly income of rideshare drivers follows the normal probability
distribution with a mean of $1,000 and a standard deviation of $100.What is
the z value of income for a driver who earns $1,100 per week? For a driver
who earns $900 per week?

What is the z-value of income for a What is the z-value of income for a
driver who earns $1,100? driver who earns $900?

x−μ $1,100−$1,000 x−μ $900−$1,000

Z= = = 1.00 Z= = = -1.00
σ $100 σ $100

Regardless of whether z is +1or -1, the area under the curve is .3413

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The Empirical Rule

To verify the Empirical Rule:

z of 1.00 = .3413 so .3413 × 2 = .6826 or about 68%

z of 2.00 = .4772 so .4772 × 2 = .9544 or about 95%
z of 3.00 = .4987 so .4987 × 2 = .9974 or about 99.7%

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The Empirical Rule Example

As part of its quality assurance program, the Autolite Battery Company

conducts tests on battery life. For a particular D-cell alkaline battery, the
mean life is 19 hours.The useful life of the battery follows a normal
distribution with a standard deviation of 1.2 hours.

1. About 68% of the batteries failed between what two values?

19 ± 1(1.2) hours;
About 68% of batteries will fail between 17.8 and 20.2 hours.
2. About 95% of the batteries failed between what two values?
19 ± 2(1.2) hours;
About 95% of batteries will fail between 16.6 and 21.4 hours.
3.Virtually all of the batteries failed between what two values?
19 ± 3(1.2) hours;
Practically all of the batteries will fail between 15.4 and 22.6 hours.

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Finding Areas under the Normal Curve
What is the z-value of income for a
driver who earns $1,100?

x−μ $1,100−$1,000
Z= = = 1.00
σ $100

Using the weekly incomes of Uber drivers:

P($1,000 < weekly income < $1,100) = .3413
P(weekly income < $1,100) = .3413 + .5000 =.8413

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Finding Areas under the Normal Curve
(2 of 4)
What is the z-value of income for a
driver who earns $790?

x−μ $790−$1,000
Z= = = -2.10
σ $100

Using the weekly incomes of Uber drivers:

P($790 <weekly income < $1,000) = .4821
P(weekly income < $790) = .5000 − .4821 = .0179

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Finding Areas Under the Normal Curve
(3 of 4)

What is the z-value of income for a

driver who earns $840?

x−μ $840−$1,000
Z= = = -1.60
σ $100

What is the z-value of income for a

driver who earns $1,200?

x−μ $1200−$1,000
Z= = = 2.00
σ $100

Using the weekly incomes of Uber drivers:

P($840 <weekly income < $1,000) = .4452
P($1,000 <weekly income < $1,200) = .4772
P($840 < weekly income < $1,200) = .4452 + .4772 = .9224

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Finding Areas Under the Normal Curve
(4 of 4)

What is the z-value of income for a

driver who earns $1,250?

x−μ $1,250−$1,000
Z= = = 2.50
σ $100

What is the z-value of income for a

driver who earns $1,150?

x−μ $1,150−$1,000
Z= = = 1.50
σ $100

Using the weekly incomes of Uber drivers:

P($1,000 <weekly income < $1,250) = .4938
P($1,000 <weekly income < $1,150) = .4332
P($1,150 < weekly income < $1,250) = .4938 − .4332 = .0606
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Finding a Value for x Using z
Layton Tire and Rubber Company wishes to set a minimum mileage
guarantee on its new MX100 tire.Tests reveal the mean mileage is
67,900 with a standard deviation of 2,050 miles and that the distribution
follows the normal distribution. Let x represent the minimum
guaranteed mileage and use the formula for z to solve so that no more
than 4% of tires need to be replaced.

z = = and from the table we find z = -1.75

so -1.75 = = therefore, x = 64,312 miles

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Approximate a Binomial Distribution
 The normal probability distribution can approximate a
binomial distribution under certain conditions
 nπ and n(1-π) must both be at least 5
 n is the number of observations
 π is the probability of a success
 The four conditions for a binomial probability distribution
are
 There are only two possible outcomes
 π (pi) remains the same from trial to trial
 The trials are independent
 The distribution results from a count of the number of
successes in a fixed number of trials
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Exponential Probability Distribution
 The exponential probability distribution describes times
between events in a sequence
 The actions occur independently at a constant rate per
unit of time or length
 It is non-negative, is positively skewed, declines steadily to
the right, and is asymptotic
 Examples of situations using the exponential distributions
 The service time for customers at the information desk
at Dallas Public Library
 The time until the next phone call arrives in a customer
service center

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The Family of Exponential Distributions
 There is not just one exponential distribution, but a family
of them
 lambda, is the rate parameter
 The lower the rate parameter, the “less skewed” the
shape of the distribution

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Exponential Distribution Formulas
 This formula describes the exponential distribution

 The area under the curve is given by this formula

 Both the mean and the standard deviation are

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Exponential Distribution Example

Orders for prescriptions arrive at a pharmacy website according to an

exponential probability distribution at a mean of one every 20 seconds.
Find the probability the next order arrives in less than 5 seconds.

P(Arrival time < 5) = 1 −

=1−
= 1 − .7788 = .2212

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Exponential Distribution Example (2 of 2)

Orders for prescriptions arrive at a pharmacy website according to an

exponential probability distribution at a mean of one every 20 seconds.
Find the probability the next order arrives in more than 40 seconds.

P(Arrival time > 40) = = .1353

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 Chapter 7 Practice Problems

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Question 1 LO-1
A uniform distribution is defined over the interval from 6 to 10.

a. What are the values for a and b?

b. What is the mean of this uniform distribution?
c. What is the standard deviation?
d. Show that the probability of any value between 6 and 10 is
equal to 1.0.
e. What is the probability that the random variable is more
than 7?
f. What is the probability that the random variable is between
7 and 9?
g. What is the probability that the random variable is equal to
7.91?

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Question 9 LO7-3

The mean of a normal probability distribution is 500; the

standard deviation is 10.

a. About 68% of the observations lie between what two

values?
b. About 95% of the observations lie between what two
values?
c. Practically all of the observations lie between what two
values?

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Question 13 LO7-3

A normal population has a mean of 20.0 and a standard

deviation of 4.0.

a. Compute the z value associated with 25.0.

b. What proportion of the population is between 20.0 and
25.0?
c. What proportion of the population is less than 18.0?

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Question 19 LO7-3

The Internal Revenue Service reported the average refund

in 2017 was $2,878 with a standard deviation of $520.
Assume the amount refunded is normally distributed.

a. What percent of the refunds are more than $3,500?

b. What percent of the refunds are more than $3,500 but
less than $4,000?
c. What percent of the refunds are more than $2,400 but
less than $4,000?

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Question 27 LO7-3

According to media research, the typical American listened to

195 hours of music in the last year. This is down from 290 hours
4 years earlier. Dick Trythall is a big country and western music
fan. He listens to music while working around the house, reading,
and riding in his truck. Assume the number of hours spent
listening to music follows a normal probability distribution with a
standard deviation of 8.5 hours.

a. If Dick is in the top 1% in terms of listening time, how many

hours did he listen last year?
b. Assume that the distribution of times 4 years earlier also
follows the normal probability distribution with a standard
deviation of 8.5 hours. How many hours did the 1% who
listen to the least music actually listen?
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Question 31 LO7-4

Waiting times to receive food after placing an order at the

local Subway sandwich shop follow an exponential
distribution with a mean of 60 seconds. Calculate the
probability a customer waits:

a. Less than 30 seconds.

b. More than 120 seconds.
c. Between 45 and 75 seconds.
d. Fifty percent of the patrons wait less than how many
seconds? What is the median?

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