© 2002 Thomson / South-Western Slide 6-1 Chapter 6 Continuous Probability Distributions.

© 2002 Thomson / South-Western Slide 6-1

Chapter 6

Continuous Probability

Distributions

© 2002 Thomson / South-Western Slide 6-2

Learning ObjectivesLearning 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 6-3

Uniform DistributionUniform Distribution

f xb a

for a x b

for

( )

1

0 all other values Area = 1

f x( )

x

1

b a

a b

The uniform distribution is a continuous distribution in which the same height, of f(X), is obtained over a range of values.

© 2002 Thomson / South-Western Slide 6-4

Example: Uniform Distribution of Lot Weights

Example: Uniform Distribution of Lot Weights

f x

for x

for

( )

1

47 4141 47

0 all other values

Area = 1

f x( )

x

1

47 41

1

6

41 47

© 2002 Thomson / South-Western Slide 6-5

Example: Uniform Distribution,continued

Mean and Standard Deviation

Example: Uniform Distribution,continued

Mean and Standard DeviationMean

=+ a b

2

Mean

=+ 41 47

2

88

244

Standard Deviation

b a

12

Standard Deviation

47 41

12

6

3 4641 732

..

© 2002 Thomson / South-Western Slide 6-6

Example: Uniform Distribution Probability, continued

Example: Uniform Distribution Probability, continued

P Xb ax x x x( )1 22 1

f x( )

x41 47

P X( )42 4545 42

47 41

1

2

42 45

45 42

47 41

1

2

Area= 0.5

© 2002 Thomson / South-Western Slide 6-7

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 6-8

Characteristics of the Normal Distribution

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.

1/2 1/2

X

© 2002 Thomson / South-Western Slide 6-9

Probability Density Function of the Normal Distribution

Probability Density Function of the Normal Distribution

f xx

Where

e

e( )

:

1

2

1

2

2

mean of X

standard deviation of X

= 3.14159 . . .

2.71828 . . .

X

© 2002 Thomson / South-Western Slide 6-10

Normal Curves for Different Means and Standard Deviations

Normal Curves for Different Means and Standard Deviations

20 30 40 50 60 70 80 90 100 110 120

5 5

10

© 2002 Thomson / South-Western Slide 6-11

Standardized Normal DistributionStandardized Normal Distribution

• A normal distribution with– a mean of zero, and – a standard deviation of

one• Z Formula

– standardizes any normal distribution

• Z Score– computed by the Z

Formula– the number of standard

deviations which a value is away from the mean

ZX

1

0

© 2002 Thomson / South-Western Slide 6-12

Z TableZ TableSecond 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 0.0000 0.0040 0.0080 0.0120 0.0160 0.0199 0.0239 0.0279 0.0319 0.03590.10 0.0398 0.0438 0.0478 0.0517 0.0557 0.0596 0.0636 0.0675 0.0714 0.07530.20 0.0793 0.0832 0.0871 0.0910 0.0948 0.0987 0.1026 0.1064 0.1103 0.11410.30 0.1179 0.1217 0.1255 0.1293 0.1331 0.1368 0.1406 0.1443 0.1480 0.1517

0.90 0.3159 0.3186 0.3212 0.3238 0.3264 0.3289 0.3315 0.3340 0.3365 0.33891.00 0.3413 0.3438 0.3461 0.3485 0.3508 0.3531 0.3554 0.3577 0.3599 0.36211.10 0.3643 0.3665 0.3686 0.3708 0.3729 0.3749 0.3770 0.3790 0.3810 0.38301.20 0.3849 0.3869 0.3888 0.3907 0.3925 0.3944 0.3962 0.3980 0.3997 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 0.4987 0.4987 0.4987 0.4988 0.4988 0.4989 0.4989 0.4989 0.4990 0.49903.40 0.4997 0.4997 0.4997 0.4997 0.4997 0.4997 0.4997 0.4997 0.4997 0.49983.50 0.4998 0.4998 0.4998 0.4998 0.4998 0.4998 0.4998 0.4998 0.4998 0.4998

© 2002 Thomson / South-Western Slide 6-13

-3 -2 -1 0 1 2 3

Table Lookup of aStandard Normal Probability

Table Lookup of aStandard Normal Probability

P Z( ) .0 1 0 3413

Z 0.00 0.01 0.02

0.00 0.0000 0.0040 0.00800.10 0.0398 0.0438 0.04780.20 0.0793 0.0832 0.0871

1.00 0.3413 0.3438 0.3461

1.10 0.3643 0.3665 0.36861.20 0.3849 0.3869 0.3888

© 2002 Thomson / South-Western Slide 6-14

Applying the Z Formula: Example, Assume….

Applying the Z Formula: Example, Assume….

X is normally distributed with = 485, and = 105 P X P Z( ) ( . ) .485 600 0 1 10 3643

For X = 485,

Z =X -

485 485

1050

For X = 600,

Z =X -

600 485

1051 10.

Z 0.00 0.01 0.02

0.00 0.0000 0.0040 0.00800.10 0.0398 0.0438 0.0478

1.00 0.3413 0.3438 0.3461

1.10 0.3643 0.3665 0.3686

1.20 0.3849 0.3869 0.3888

© 2002 Thomson / South-Western Slide 6-15

Normal Approximation of the Binomial Distribution

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 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 6-16

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 of the normal distribution.

© 2002 Thomson / South-Western Slide 6-17

• Conversion equations

• Conversion example:

Normal Approximation of Binomial: Parameter Conversion

Normal Approximation of Binomial: Parameter Conversion

n p

n p q

Given that X has a binomial distribution, find

and P X n p

n p

n p q

( | . ).

( )(. )

( )(. )(. ) .

25 60 30

60 30 18

60 30 70 3 55

© 2002 Thomson / South-Western Slide 6-18

Normal Approximation of Binomial: Interval Check

Normal Approximation of Binomial: Interval Check

3 18 3 355 18 10 65

3 7 35

3 28 65

( . ) .

.

.

0 10 20 30 40 50 60n

70

© 2002 Thomson / South-Western Slide 6-19

Normal Approximation of Binomial: Correcting for Continuity

Normal Approximation of Binomial: Correcting for Continuity

Values Being

DeterminedCorrection

XXXXXX

+.50-.50-.50+.05

-.50 and +.50+.50 and -.50

The binomial probability,

and

is approximated by the normal probability

P(X 24.5| and

P X n p( | . )

. ).

25 60 30

18 3 55

© 2002 Thomson / South-Western Slide 6-20

Normal Approximation of Binomial: Graphs

Normal Approximation of Binomial: Graphs

0

0.02

0.04

0.06

0.08

0.10

0.12

6 8 10 12 14 16 18 20 22 24 26 28 30

© 2002 Thomson / South-Western Slide 6-21

Normal Approximation of Binomial: Computations

Normal Approximation of Binomial: Computations

252627282930313233

Total

0.01670.00960.00520.00260.00120.00050.00020.00010.00000.0361

X P(X)

The normal approximation,

P(X 24.5| and

18 355

24 5 18

355

183

5 0 183

5 4664

0336

. )

.

.

( . )

. .

. .

.

P Z

P Z

P Z

© 2002 Thomson / South-Western Slide 6-22

Exponential DistributionExponential 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

f X XXe( ) ,

for 0 0

© 2002 Thomson / South-Western Slide 6-23

Graphs of Selected Exponential Distributions

Graphs of Selected Exponential Distributions

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

0 1 2 3 4 5 6 7 8

© 2002 Thomson / South-Western Slide 6-24

Exponential Distribution Example:Probability Computation

Exponential Distribution Example:Probability Computation

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 1 2 3 4 5

P X X

X

P X

ee

00

2 1212 2

0907

| .( . )( )

.