5_engineering_probability_and_statistics.pdf

7/29/2019 5_engineering_probability_and_statistics.pdf

1/90

ENGINEERING PROBABILITY AND STATISTICS

DISPERSION, MEAN, MEDIAN, AND MODE

VALUES

IfX1,X2, ,Xnrepresent the values of a random sample ofnitems or observations, the arithmetic mean of these items or

observations, denoted X, is dened as

.

X n X X X n X

X n

1 1

for sufficiently large values of

n ii

n

1 21

"

f= + + + =

n

=_ _ _i i i !

The weighted arithmetic mean is

,Xw

w Xwhere

wi

i i= !

!

Xi= the value of the ithobservation, and

wi= the weight applied toXi.

The variance of the population is the arithmetic mean of

thesquared deviations from the population mean. If isthe arithmetic mean of a discrete population of sizeN, the

population variance is dened by

/

/

N X X X

N X

1

1

N

ii

N

21

2

2

2 2

2

1

f= - + - + + -

= -

v n n n

n=

^ ^ ^ _^ _

h h h ih i

9 C!

Thestandard deviation of the population is

/N X1 i2

= -v n^ _h i!

Thesample variance is

/s n X X1 1 ii

n2

2

1= - -

=

^ `h j7 A !Thesample standard deviation is

/s n X X1 1 ii

n 2

1= - -

=

^ `h j7 A !Thesamplecoefcientofvariation= /CV s X =

Thesample geometric mean = X X X Xnn 1 2 3 f

Thesample root-mean-square value = /n X1 i2^ h !

When the discrete data are rearranged in increasing order and

n is odd, the median is the value of the n2

1th

+b l itemWhen n is even, the median is the average of the

and .n n2 2

1 itemsth th

+b bl lThe mode of a set of data is the value that occurs with

greatest frequency.

Thesample range R is the largest sample value minus the

smallest sample value.

PERMUTATIONS AND COMBINATIONS

Apermutation is a particular sequence of a given set of

objects. A combination is the set itself without reference to

order.

1. The number of differentpermutations ofn distinct objects

taken r at a time is

,!

!P n rn r

n=-^ ^h h

nPris an alternative notation forP(n,r)

2. The number of different combinations ofn distinct objects

taken r at a time is

,!

,

! !

!C n rr

P n r

r n r

n= =

-

^ ^^

h hh7 A

nCrandn

re oare alternative notations forC(n,r)

3. The number of differentpermutations ofn objects taken

n at a time, given that niare of type i, where i = 1, 2, , kand ni= n, is

; , , ,! ! !

!P n n n nn n n

nk

k1 2

1 2f

f=_ i

SETS

DeMorgans Law

A B A B

A B A B

, +

+ ,

=

=

Associative Law

A B C A B C

A B C A B C

, , , ,

+ + + +

=

=

^ ]^ ]

h gh g

Distributive Law

A B C A B A C

A B C A B A C

, + , + ,

+ , + , +

=

=

^ ] ^^ ] ^

h g hh g h

LAWS OF PROBABILITY

Property 1. General Character of Probability

The probabilityP(E) of an eventEis a real number in the

range of 0 to 1. The probability of an impossible event is 0 and

that of an event certain to occur is 1.

Property 2. Law of Total Probability

P(A + B) = P(A) + P(B) P(A, B), where

P(A +B) = the probability that eitherA orB occur alone or

that both occur together,

P(A) = the probability thatA occurs,

P(B) = the probability thatB occurs, and

P(A,B) = the probability that bothA andB occur

simultaneously.



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1ENGINEERING PROBABILITY AND STATISTICS

Property 3. Law of Compound or Joint Probability

If neitherP(A) norP(B) is zero,

P(A, B) = P(A)P(B |A) = P(B)P(A |B), where

P(B |A) = the probability thatB occurs given the fact thatAhas occurred, and

P(A |B) = the probability thatA occurs given the fact thatBhas occurred.

If eitherP(A) orP(B) is zero, thenP(A,B) = 0.

Bayes Theorem

A

P B AP A B P B

P B P A B

P A

A

P B B

B

where is the probability of event within the

population of

is the probability of event within the

population of

j

j

i ii

n

j j

j

j j

1

=

=

__ _

_ _

_

_

ii i

i i

i

i

!

PROBABILITY FUNCTIONS

A random variableXhas a probability associated with each

of its possible values. The probability is termed a discrete

probability ifXcan assume only discrete values, or

X = x1, x2, x3, , xn

The discrete probability of any single event,X=xi, occurring

is dened asP(xi) while theprobability mass function of the

random variableXis dened by

f(xk) =P(X= x

k), k= 1, 2, ..., n

Probability Density Function

IfXis continuous, theprobability density function, f, isdened such that

P a X b f x dxa

b

# # =^ ^h h#

Cumulative Distribution Functions

The cumulative distribution function, F, of a discrete random

variableXthat has a probability distribution described byP(xi)

is dened as

, , , ,F x P x P X x m n1 2m kk

m

m1

f#= = ==

_ _ _i i i!

IfXis continuous, the cumulative distribution function, F, is

dened by

F x f t dtx

=

3-

^ ^h h#

which implies thatF(a) is the probability thatX a.

Expected Values

LetXbe a discrete random variable having a probability

mass function

f(xk), k= 1, 2,..., n

The expected value ofXis dened as

E X x f xkk

n

k1

= =n=

_ i6 @ !

The variance ofXis dened as

V X x f xkk

n

k2 2

1= = -v n

=

_ _i i6 @ !

LetXbe a continuous random variable having a density

functionf(X) and let Y=g(X) be some general function.

The expected value ofYis:

E Y E g X g x f x dx= =3

3

-

] ^ ^g h h6 7@ A #The mean or expected value of the random variableXis now

dened as

E X xf x dx= =n3

3

-

^ h6 @ #

while the variance is given by

V X E X x f x dx22 2

= = - = -v n n

3

3

-

^ ^ ^h h h6 9@ C #The standard deviation is given by

V X=v 6 @

The coefcient of variation is dened as /.

Sums of Random Variables

Y = a1X1+ a2X2+ + anXn

The expected value ofYis:

E Y a E X a E X a E Xy n n1 1 2 2 f= = + + +n ] ^ ^ _g h h iIf the random variables are statistically independent, then the

variance ofYis:

V Y a V X a V X a V X

a a a

y n n

n n

212

1 22

22

12

12

22

22 2 2

f

f

= = + + +

= + + +

v

v v v

] ^ ^ _g h h i

Also, the standard deviation ofYis:

y y2

=v v


3/9


Binomial Distribution

P(x) is the probability thatx successes will occur in n trials.

Ifp = probability of success and q = probability of failure =

1 p, then

,! !

! ,P x C n x p qx n x

n p qnx n x x n x

= =-

- -^ ^^

h hh

where

x = 0, 1, 2,, n,

C(n, x) = the number of combinations, andn, p =parameters.

Normal Distribution (Gaussian Distribution)

This is a unimodal distribution, the mode beingx = , withtwo points of inection (each located at a distance to eitherside of the mode). The averages ofn observations tend to

become normally distributed as n increases. The variatex is

said to be normally distributed if its density functionf(x) is

given by an expression of the form

, wheref x e2

1 x21

2

=v r

--

v

n

^ ch m

= the population mean, = the standard deviation of the population, and

x3 3# #-

When = 0 and 2= = 1, the distribution is called astandardizedorunit normaldistribution. Then

, where .f x e x21 /x 22

3 3# #= -r

-^ h

It is noted that Zx

=-

v

nfollows a standardized normal

distribution function.

A unit normal distribution table is included at the end of this

section. In the table, the following notations are utilized:

F(x) = the area under the curve from tox,R(x) = the area under the curve fromx to , andW(x) = the area under the curve between x andx.

The Central Limit Theorem

LetX1,X2, ...,Xnbe a sequence of independent and identically

distributed random variables each having mean and variance2. Then for large n, the Central Limit Theorem asserts thatthe sum

Y=X1 +X2 + ... Xnis approximately normal.

y =n n

and the standard deviation

ny =v

v

t-Distribution

The variate tis dened as the quotient of two independent

variatesx and rwherex is unit normaland r is the root mean

square ofn other independent unit normal variates; that is,

t=x/r. The following is the t-distribution with n degrees of

freedom:

/

/

/

f tn n

n

t n2

1 2

1

1/n

21 2

=+

+rC

C

+

^^

^

_^

hh

h

ih

7 A

where t .

A table at the end of this section gives the values oft, n for

values of and n. Note that in view of the symmetry of thet-distribution, t1,n= t,n.

The function for follows:

f t dtt , n

=a3

a

^ h#

2- DistributionIfZ1,Z2, ...,Znare independent unit normal random

variables, then

Z Z Zn2

12

22 2

f= + + +|

is said to have a chi-square distribution with n degrees of

freedom. The density function is shown as follows:

, >fn

e x

x

2

21

20

x n

22

2

21

2

2

=|

C

--_

bci

lm

A table at the end of this section gives values of , n2

|a

for

selected values ofand n.

Gamma Function, >n t e dt n 0n t1

0=C

3 - -^ h #

LINEAR REGRESSION

Least Squares

, where

: ,

and slope: / ,

/ ,

/ ,

sample size,

/ , and

/ .

y

y a bx

a y bx

b S S

S x y n x y

S x n x

n

y n y

x n x

1

1

1

1

intercept

xy xx

xy ii

n

i ii

n

ii

n

xx ii

n

ii

n

ii

n

ii

n

1 1 1

2

1 1

2

1

1

= +

= -

=

= -

= -

=

=

=

-

= = =

= =

=

=

t t

t t

t

^ d d^ d

^ d^ d

h n nh n

h nh n

! ! !

! !

!

!


4/9


Standard Error of Estimate

, where

/

SS n

S S SMSE

S y n y

2

1

exx

xx yy xy

yy ii

n

ii

n

2

2

2

1 1

2

=-

-

=

= -= =

^

^ dh

h n! !

Condence Interval for a

a t n Sx

MSE1

/ , nxx

2 2

2

! +-at e oCondence Interval for b

b tS

MSE/ , n

xx2 2! -a

t

Sample Correlation Coefcient

RS S

S

RS S

S

xx yy

xy

xx yy

xy22

=

=

HYPOTHESIS TESTING

Consider an unknown parameterof a statistical distribution. Let the null hypothesis beH0: = 0

and let the alternative hypothesis be

H1: = 1

RejectingH0when it is true is known as a type I error, while acceptingH0when it is wrong is known as a type II error.

Furthermore, the probabilities of type I and type II errors are usually represented by the symbols and , respectively: =probability (type I error)

=probability (type II error)

The probability of a type I error is known as the level of signicance of the test.

Assume that the values ofand are given. The sample size can be obtained from the following relationships. In (A) and (B),1is the value assumed to be the true mean.

A : ; :

/ /Z Z

H H

n n/ /

0 0 1 0

02

02

!=

=-

+ --

-

n n n n

bv

n n

v

n nU Ua a

]

e eg

o oAn approximate result is

Z Zn

/

1 0

2

22 2

-

-

+

n n

va b

_

^i

h

B : ; : >

/Z

Z

H H

n

nZ

0 0 1 0

0

1 0

2

2 2

=

=-

+

=

-

+

n n n n

bv

n n

n n

v

U a

a b

e

_

_

]

o

i

i

g

Refer to the Hypothesis Testing table in the INDUSTRIAL ENGINEERING section of this handbook.


5/9


CONFIDENCE INTERVALS

Condence Interval for the Mean of a Normal Distribution

A Standard deviation is known

B Standard deviation is not known

where t corresponds to 1 degrees of freedom.n

X Zn

X Zn

X t

n

s X t

n

s

/ /

/ /

/2

2 2

2 2

# #

# #

- +

- +

-

v

vn

v

v

n

a a

a a

a

]

]

g

g

Condence Interval for the Difference Between Two Means 1and 2(A) Standard deviations 1and 2known

X X Zn n

X X Zn n/ /1 2 2 1

12

2

22

1 2 1 2 21

12

2

22

# #- - + - - + +v v

n nv v

a a

(B) Standard deviations 1and 2are not known

where t corresponds to 2 degrees of freedom.n n

X X t n n

n nn S n S

X X t n n

n nn S n S

2

1 1 1 1

2

1 1 1 1

/ /

/2 1 2

1 2 2 1 2

1 21 1

22 2

2

1 2 1 2 2 1 2

1 21 1

22 2

2

# #- -+ -

+ - + -

- - ++ -

+ - + -

+ -

n na a

a

c ^ ^ c ^ ^m h h m h h8 8B B

Condence Intervals for the Variance 2of a Normal Distribution

x

n s

x

n s1 1

/ , / ,n n2 12

22

1 2 12

2

# #- -

v

- - -a a

^ ^h h

Sample Size

zn

Xn

x

z 22

=-

=-v

n

n

va= G


6/9


UNIT NORMAL DISTRIBUTION

x f(x) F(x) R(x) 2R(x) W(x)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2.0

2.1

2.2

2.3

2.4

2.5

2.6

2.7

2.8

2.9

3.0

Fractiles

1.2816

1.6449

1.9600

2.0537

2.3263

2.5758

0.3989

0.3970

0.3910

0.3814

0.3683

0.3521

0.3332

0.3123

0.2897

0.2661

0.2420

0.2179

0.1942

0.1714

0.1497

0.1295

0.1109

0.0940

0.0790

0.0656

0.0540

0.0440

0.0355

0.0283

0.0224

0.0175

0.0136

0.0104

0.0079

0.0060

0.0044

0.1755

0.1031

0.0584

0.0484

0.0267

0.0145

0.5000

0.5398

0.5793

0.6179

0.6554

0.6915

0.7257

0.7580

0.7881

0.8159

0.8413

0.8643

0.8849

0.9032

0.9192

0.9332

0.9452

0.9554

0.9641

0.9713

0.9772

0.9821

0.9861

0.9893

0.9918

0.9938

0.9953

0.9965

0.9974

0.9981

0.9987

0.9000

0.9500

0.9750

0.9800

0.9900

0.9950

0.5000

0.4602

0.4207

0.3821

0.3446

0.3085

0.2743

0.2420

0.2119

0.1841

0.1587

0.1357

0.1151

0.0968

0.0808

0.0668

0.0548

0.0446

0.0359

0.0287

0.0228

0.0179

0.0139

0.0107

0.0082

0.0062

0.0047

0.0035

0.0026

0.0019

0.0013

0.1000

0.0500

0.0250

0.0200

0.0100

0.0050

1.0000

0.9203

0.8415

0.7642

0.6892

0.6171

0.5485

0.4839

0.4237

0.3681

0.3173

0.2713

0.2301

0.1936

0.1615

0.1336

0.1096

0.0891

0.0719

0.0574

0.0455

0.0357

0.0278

0.0214

0.0164

0.0124

0.0093

0.0069

0.0051

0.0037

0.0027

0.2000

0.1000

0.0500

0.0400

0.0200

0.0100

0.0000

0.0797

0.1585

0.2358

0.3108

0.3829

0.4515

0.5161

0.5763

0.6319

0.6827

0.7287

0.7699

0.8064

0.8385

0.8664

0.8904

0.9109

0.9281

0.9426

0.9545

0.9643

0.9722

0.9786

0.9836

0.9876

0.9907

0.9931

0.9949

0.9963

0.9973

0.8000

0.9000

0.9500

0.9600

0.9800

0.9900


7/9


STUDENT'S t-DISTRIBUTION

VALUES OF

,n

t

t,n

df

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

1819

20

21

22

23

24

25

26

27

28

29

30

0.10

3.078

1.886

1.638

1.533

1.476

1.440

1.415

1.397

1.383

1.372

1.363

1.356

1.350

1.345

1.341

1.337

1.333

1.330

1.328

1.325

1.323

1.321

1.319

1.318

1.316

1.315

1.314

1.313

1.311

1.282

1.310

0.05

6.314

2.920

2.353

2.132

2.015

1.943

1.895

1.860

1.833

1.812

1.796

1.782

1.771

1.761

1.753

1.746

1.740

1.734

1.729

1.725

1.721

1.717

1.714

1.711

1.708

1.706

1.703

1.701

1.699

1.645

1.697

0.025

12.706

4.303

3.182

2.776

2.571

2.447

2.365

2.306

2.262

2.228

2.201

2.179

2.160

2.145

2.131

2.120

2.110

2.101

2.093

2.086

2.080

2.074

2.069

2.064

2.060

2.056

2.052

2.048

2.045

1.960

2.042

0.01

31.821

6.965

4.541

3.747

3.365

3.143

2.998

2.896

2.821

2.764

2.718

2.681

2.650

2.624

2.602

2.583

2.567

2.552

2.539

2.528

2.518

2.508

2.500

2.492

2.485

2.479

2.473

2.467

2.462

2.326

2.457

df

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

1819

20

21

22

23

24

25

26

27

28

29

30

0.15

1.963

1.386

1.350

1.190

1.156

1.134

1.119

1.108

1.100

1.093

1.088

1.083

1.079

1.076

1.074

1.071

1.069

1.067

1.066

1.064

1.063

1.061

1.060

1.059

1.058

1.058

1.057

1.056

1.055

1.036

1.055

0.20

1.376

1.061

0.978

0.941

0.920

0.906

0.896

0.889

0.883

0.879

0.876

0.873

0.870

0.868

0.866

0.865

0.863

0.862

0.861

0.860

0.859

0.858

0.858

0.857

0.856

0.856

0.855

0.855

0.854

0.842

0.854

0.25

1.000

0.816

0.765

0.741

0.727

0.718

0.711

0.706

0.703

0.700

0.697

0.695

0.694

0.692

0.691

0.690

0.689

0.688

0.688

0.687

0.686

0.686

0.685

0.685

0.684

0.684

0.684

0.683

0.683

0.674

0.683

0.005

63.657

9.925

5.841

4.604

4.032

3.707

3.499

3.355

3.250

3.169

3.106

3.055

3.012

2.977

2.947

2.921

2.898

2.878

2.861

2.845

2.831

2.819

2.807

2.797

2.787

2.779

2.771

2.763

2.756

2.576

2.750


8/9


9/9

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5_engineering_probability_and_statistics.pdf