continuous probability distributions -...

Continuous Probability DistributionsChapter 7

Copyright © 2011 by the McGraw-Hill Companies, Inc. All rights reserved.McGraw-Hill/Irwin

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제6장 연속확률분포와정규분포

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LEARNING OBJECTIVESLO1. Understand the difference between discrete and continuous

distributions. LO2. Compute the mean and the standard deviation for a uniform

distribution.LO3. Compute probabilities by using the uniform distribution.LO4. List the characteristics of the normal probability distribution.LO5. Define and calculate z values.LO6. Determine the probability an observation is between two points on

a normal probability distribution.LO7. Determine the probability an observation is above (or below) a

point on a normal probability distribution.

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Types of Random Variables

DISCRETE RANDOM VARIABLE A random variable that can assume only certain clearly separated values. It is usually the result of counting something.

CONTINUOUS RANDOM VARIABLE can assume an infinite number of values within a given range. It is usually the result of some type of measurement

Learning Objective 1 Distinguish betweendiscrete and continuousprobability distributions.

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Discrete Random Variables

EXAMPLES1. The number of students in a class.2. The number of children in a family.3. The number of cars entering a carwash in a hour.4. Number of home mortgages approved by Coastal Federal

Bank last week.

DISCRETE RANDOM VARIABLE A random variable that can assume only certain clearly separated values. It is usually the result of counting something.

LO1

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Continuous Random Variables

EXAMPLES The length of each song on the latest Tim McGraw album. The weight of each student in this class. The temperature outside as you are reading this book. The amount of money earned by each of the more than 750

players currently on Major League Baseball team rosters.

CONTINUOUS RANDOM VARIABLE can assume an infinite number of values within a given range. It is usually the result of some type of measurement

LO1

The Uniform Distribution

The uniform probability distribution is perhaps the simplest distributionfor a continuous random variable.

This distribution is rectangular in shapeand is defined by minimum and maximum values.

LO1

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The Uniform Distribution Mean and Standard Deviation

Learning Objective 2 Compute the mean andthe standard deviation for auniform probabilitydistribution.

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The Uniform Distribution An ExampleSouthwest 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.

1. Draw a graph of this distribution.2. Show that the area of this uniform distribution is 1.00.3. How long will a student “typically” have to wait for a bus? In other words

what is the mean waiting time? What is the standard deviation of the waiting times?

4. What is the probability a student will wait more than 25 minutes5. What is the probability a student will wait between 10 and 20 minutes?

Learning Objective 3 Compute probabilities byusing the uniform probabilitydistribution.

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The Uniform Distribution - Example

1. Draw a graph of this distribution.

LO3

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2. Show that the area of this distribution is 1.00

LO3

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3. How long will a student “typically” have to wait for a bus? In other words what is the mean waiting time?

What is the standard deviation of the waiting times?

LO3


4. What is the probability a student will wait more than 25 minutes? 0.1667

)5()030(

1

ase)(height)(b30)TimeWait 25(

P

LO3

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5. What is the probability a student will wait between 10 and 20 minutes? 0.3333

ase)(height)(b20)Time Wait

)10()030(

110(P

LO3

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Characteristics of a Normal Probability Distribution

1. It is bell-shaped and has a single peak at the center of the distribution.

2. It is symmetrical about the mean3. It is asymptotic: The curve gets closer and closer to the X-axis

but never actually touches it. 4. The location of a normal distribution is determined by the

mean,, the dispersion or spread of the distribution is determined by the standard deviation,σ .

5. The arithmetic mean, median, and mode are equal6. The total area under the curve is 1.00; half the area under the

normal curve is to the right of this center point and the other half to the left of it

Learning Objective 4 List the characteristics ofthe normal probabilitydistribution.

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The Normal Distribution - Graphically

LO4

The Family of Normal Distribution

Different Means and Standard Deviations

Equal Means and Different Standard Deviations

Different Means and Equal Standard Deviations

LO4

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The Standard Normal Probability Distribution The standard normal distribution is a normal

distribution with a mean of 0 and a standard deviation of 1.

It is also called the z distribution. A z-value is the signed distance between a

selected value, designated X, and the population mean , divided by the population standard deviation, σ.

The formula is:

Learning Objective 5 Define and calculate zvalues.

Areas Under the Normal CurveLO5

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The Normal Distribution – Example

The weekly incomes of shift foremen in the glass industry follow the normal probability distribution with a mean of $1,000 and a standard deviation of $100.

What is the z value for the income, let’s call it X, of a foreman who earns $1,100 per week? For a foreman who earns $900 per week?

Learning Objective 6 Determine the probabilityan observation is between two points on a normal probability distribution.

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The Empirical Rule

About 68 percent of the area under the normal curve is within one standard deviation of the mean.

About 95 percent is within two standard deviations of the mean.

Practically all is within three standard deviationsof the mean.

LO6

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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.

Answer the following questions.1. About 68 percent of the

batteries failed between what two values?

2. About 95 percent of the batteries failed between what two values?

3. Virtually all of the batteries failed between what two values?

LO6

Normal Distribution – Finding Probabilities

In an earlier example we reported that the mean weekly income of a shift foreman in the glass industry is normally distributed with a mean of $1,000 and a standard deviation of $100.

What is the likelihood of selecting a foreman whose weekly income is between $1,000 and $1,100?

LO6

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Normal Distribution – Finding Probabilities

LO6

Finding Areas for Z Using Excel

The Excel function=NORMDIST(x,Mean,Standard_dev,Cumu)=NORMDIST(1100,1000,100,true)generates area (probability) fromZ=1 and below

LO6

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Normal Distribution – Finding Probabilities (Example 2)

Refer to the information regarding the weekly income of shift foremen in the glass industry. The distribution of weekly incomes follows the normal probability distribution with a mean of $1,000 and a standard deviation of $100.

What is the probability of selecting a shift foreman in the glass industry whose income is:

Between $790 and $1,000?

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Less than $790?

LO6

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Between $840 and $1,200?

LO6

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Between $1,150 and $1,250

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Using Z in Finding X Given Area Example

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 2,050miles and that the distribution of miles follows the normal probability distribution. Layton wants to set the minimum guaranteed mileage so that no more than 4 percent of the tires will have to be replaced.

What minimum guaranteed mileage should Layton announce?

Learning Objective 7 Determine the probabilityan observation is above (or below) a normal probability distribution.

Using Z in Finding X Given Area - Example

64,312x

)1.75(2,050-67,900x

67,900-x)1.75(2,050-

for x solving then ,2,05067,900-x1.75-

:equation theinto ngsubstitutiThen 1.75.- of alue z a gives which0.4599, is 0.4600 closest to area theB.1,Appendix Using

0.0400-0.5000by found 0.4600, is x and 67,900between area Theninformatio 4% theusing found is z of valueThe

050,2900,67-z

:formula theusing X Solve

xx

LO7

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Using Z in Finding X Given Area - ExcelLO6

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