1.1 Properties of Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Contents Prefacev

• Probability1

1.1 Properties of Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Methods of Enumeration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Conditional Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.4 Independent Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.5 Bayes' Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

• Discrete Distributions7

2.1 Random Variables of the Discrete Type . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 Mathematical Expectation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.3 Special Mathematical Expectations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.4 The Binomial Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.5 The Negative Binomial Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.6 The Poisson Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

• Continuous Distributions19

3.1 Random Variables of the Continuous Type . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.2 The Exponential, Gamma, and Chi-Square Distributions . . . . . . . . . . . . . . . . . 25 3.3 The Normal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.4 Additional Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

• Bivariate Distributions33

4.1 Bivariate Distributions of the Discrete Type . . . . . . . . . . . . . . . . . . . . . . . . 33 4.2 The Correlation Coe±cient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 4.3 Conditional Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 4.4 Bivariate Distributions of the Continuous Type . . . . . . . . . . . . . . . . . . . . . . 36 4.5 The Bivariate Normal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

• Distributions of Functions of Random Variables43

5.1 Functions of One Random Variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 5.2 Transformations of Two Random Variables . . . . . . . . . . . . . . . . . . . . . . . . 44 5.3 Several Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 5.4 The Moment-Generating Function Technique . . . . . . . . . . . . . . . . . . . . . . . 50 5.5 Random Functions Associated with Normal Distributions . . . . . . . . . . . . . . . . 52 5.6 The Central Limit Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 5.7 Approximations for Discrete Distributions . . . . . . . . . . . . . . . . . . . . . . . . . 56 5.8 Chebyshev's Inequality and Convergence in Probability . . . . . . . . . . . . . . . . . 58 5.9 Limiting Moment-Generating Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 59 Copyrightc°2015 Pearson Education, Inc.Copyrightc°2015 Pearson Education, Inc.Copyrightc°2015 Pearson Education, Inc.Probability and Statistical Inference 9e Robert Hogg Elliot Tanis Dale Zimmerman (Solutions Manual All Chapters, 100% Original Verified, A+ Grade) 1 / 4

ivContents

• Point Estimation61

6.1 Descriptive Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 6.2 Exploratory Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 6.3 Order Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 6.4 Maximum Likelihood Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 6.5 A Simple Regression Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 6.6 Asymptotic Distributions of Maximum Likelihood Estimators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 6.7 Su±cient Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 6.8 Bayesian Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 6.9 More Bayesian Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

• Interval Estimation85

7.1 Con¯dence Intervals for Means . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 7.2 Con¯dence Intervals for the Di®erence of Two Means . . . . . . . . . . . . . . . . . . . 86 7.3 Con¯dence Intervals For Proportions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 7.4 Sample Size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 7.5 Distribution-Free Con¯dence Intervals for Percentiles . . . . . . . . . . . . . . . . . . . 90 7.6 More Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 7.7 Resampling Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

• Tests of Statistical Hypotheses105

8.1 Tests About One Mean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 8.2 Tests of the Equality of Two Means . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 8.3 Tests about Proportions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 8.4 The Wilcoxon Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 8.5 Power of a Statistical Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 8.6 Best Critical Regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 8.7 Likelihood Ratio Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

• More Tests125

9.1 Chi-Square Goodness-of-Fit Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 9.2 Contingency Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 9.3 One-Factor Analysis of Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 9.4 Two-Way Analysis of Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 9.5 General Factorial and 2 k Factorial Designs . . . . . . . . . . . . . . . . . . . . . . . . . 133 9.6 Tests Concerning Regression and Correlation . . . . . . . . . . . . . . . . . . . . . . . 134 9.7 Statistical Quality Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 Copyrightc°2015 Pearson Education, Inc.Copyrightc°2015 Pearson Education, Inc.Copyrightc°2015 Pearson Education, Inc. 2 / 4

Chapter 1 Probability1 Chapter 1 Probability 1.1 Properties of Probability 1.1-2Sketch a ¯gure and ¯ll in the probabilities of each of the disjoint sets.

LetA=finsure more than one carg,P(A) = 0:85.

LetB=finsure a sports carg,P(B) = 0:23.

LetC=finsure exactly one carg,P(C) = 0:15.

It is also given thatP(A\B) = 0:17. Since A\C=Á,P(A\C) = 0. It follows that

P(A\B\C

) = 0:17. Thus P(A

\B\C

) = 0:06 and P(A

\B

\C) = 0:09.

1.1-4 (a)S=fHHHH; HHHT; HHTH; HTHH; THHH; HHTT; HTTH; TTHH; HTHT; THTH; THHT; HTTT; THTT; TTHT; TTTH; TTTTg; (b) (i)5/16,(ii)0,(iii)11/16,(iv)4/16,(v)4/16,(vi)9/16,(vii)4/16.

1.1-6 (a)P(A[B) = 0:4 + 0:5 ¡0:3 = 0:6; (b) A = (A \B

)[(A\B)

P(A) = P(A\B

) +P(A\B)

0:4 = P(A\B

) + 0:3

P(A\B

) = 0:1;

(c)P(A

[B

) =P[(A\B)

] = 1¡P(A\B) = 1¡0:3 = 0:7.

1.1-8LetA=flab work doneg,B=freferral to a specialistg,

P(A) = 0:41; P (B) = 0:53; P ([A[B]

) = 0:21.

P(A[B) =P(A) +P(B)¡P(A\B)

0:79 = 0:41 + 0:53¡P(A\B)

P(A\B) = 0:41 + 0:53¡0:79 = 0:15:

1.1-10A[B[C =A[(B[C)

P(A[B[C) =P(A) +P(B[C)¡P[A\(B[C)]

=P(A) +P(B) +P(C)¡P(B\C)¡P[(A\B)[(A\C)]

=P(A) +P(B) +P(C)¡P(B\C)¡P(A\B)¡P(A\C)

+P(A\B\C):

1.1-12 (a)1/3;(b)2/3;(c)0;(d)1/2.Copyrightc°2015 Pearson Education, Inc.Copyrightc°2015 Pearson Education, Inc.Copyrightc°2015 Pearson Education, Inc. 3 / 4

2Section1.2MethodsofEnumeration

1.1-14P(A)=

2[r¡r( p 3=2)] 2r =1¡ p 3 2

:

1.1-16Notethattherespectiveprobabilitiesarep0;p1=p0=4;p2=p0=4 2

;¢¢¢.

1 X k=0 p0 4 k =1 p0

1¡1=4

=1 p0 = 3 4 1¡p0¡p1=1¡ 15 16 = 1 16 .

1.2MethodsofEnumeration 1.2-2(a)(4)(5)(2)=40;(b)(2)(2)(2)=8.

1.2-4(a)4 µ 6 3 ¶ =80; (b)4(2 6

)=256;

(c)

(4¡1+3)!

(4¡1)!3!

=20.

1.2-6S=fDDD,DDFD,DFDD,FDDD,DDFFD,DFDFD,FDDFD,DFFDD,

FDFDD,FFDDD,FFF,FFDF,FDFF,DFFFFFDDF,FDFDF,

DFFDF,FDDFF,DFDFF,DDFFFgsothereare20possibilities.

1.2-83¢3¢2

12

=36;864:

1.2-10

µ n¡1 r ¶ + µ n¡1 r¡1 ¶ = (n¡1)!r!(n¡1¡r)!+ (n¡1)!(r¡1)!(n¡r)!= (n¡r)(n¡1)!+r(n¡1)!r!(n¡r)!= n!r!(n¡r)!= µ n r ¶

:

1.2-120=(1¡1)

n = n X r=0 µ n r ¶ (¡1) r (1) n¡r = n X r=0 (¡1) r µ n r ¶

:

2 n

=(1+1)

n = n X r=0 µ n r ¶ (1) r (1) n¡r = n X r=0 µ n r ¶

:

1.2-14

µ

10¡1+36

36 ¶ = 45!36!9!

=886;163;135.

1.2-16(a) µ 19 3 ¶µ

52¡19

6 ¶ µ 52 9 ¶ =

102;486

351;325

=0:2917;

(b) µ 19 3 ¶µ 10 2 ¶µ 7 1 ¶µ 3

¶µ 5 1 ¶µ 2

¶µ 6 2 ¶ µ 52 9 ¶ = 7;695

1;236;664

=0:00622:

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