Instructor's Solutions Manual
FUNDAMENTALS
OF PROBABILITY
WITH STOCHASTIC PROCESSES
FOURTH EDITION
SAEED GHAHRAMANI
Western New England University
Springfield, Massachusetts, USA
A CHAPMAN & HALL BOOK
, C ontents
1 Axioms of Probability 1
1.2 Sample Space and Events 1
1.4 Basic Theorems 4
1.7 Random Selection of Points from Intervals 10
Review Problems 13
Companion for Chapter 1 19
1B Applications of Probability to Genetics 19
2 Combinatorial Methods 21
2.2 Counting Principle 21
2.3 Permutations 24
2.4 Combinations 27
2.5 Stirling’ Formula 42
Review Problems 42
3 Conditional Probability and Independence 47
3.1 Conditional Probability 47
3.2 The Multiplication Rule 52
3.3 Law of Total Probability 55
3.4 Bayes’ Formula 60
3.5 Independence 66
Review Problems 76
Companion for Chapter 3 80
3B More on Applications of Probability to Genetics 80
, Contents iii
4
Distribution Functions and
Discrete Random Variables
85
4.2 Distribution Functions 85
4.3 Discrete Random Variables 89
4.4 Expectations of Discrete Random Variables 94
4.5 Variances and Moments of Discrete Random Variables 100
4.6 Standardized Random Variables 107
Review Problems 107
5 Special Discrete Distributions 110
5.1 Bernoulli and Binomial Random Variables 110
5.2 Poisson Random Variable 117
5.3 Other Discrete Random Variables 125
Review Problems 132
6 Continuous Random Variables 137
6.1 Probability Density Functions 137
6.2 Density Function of a Function of a Random Variable 141
6.3 Expectations and Variances 144
Review Problems 151
7 Special Continuous Distributions 153
7.1 Uniform Random Variable 153
7.2 Normal Random Variable 158
7.3 Exponential Random Variables 166
7.4 Gamma Distribution 171
7.5 Beta Distribution 175
7.6 Survival Analysis and Hazard Function 180
Review Problems 183
8 Bivariate Distributions 187
8.1 Joint Distribution of Two Random Variables 187
8.2 Independent Random Variables 200
8.3 Conditional Distributions 209
8.4 Transformations of Two Random Variables 218
Review Problems 230
, Contents iv
9 Multivariate Distributions 238
9.1 Joint Distribution of n > 2 Random Variables 238
9.2 Order Statistics 248
9.3 Multinomial Distributions 253
Review Problems 255
10 More Expectations and Variances 260
10.1 Expected Values of Sums of Random Variables 260
10.2 Covariance 265
10.3 Correlation 274
10.4 Conditioning on Random Variables 276
10.5 Bivariate Normal Distribution 289
Review Problems 292
Companion for Chapter 10 299
10B Pattern Appearance 299
11
Sums of Independent Random 300
Variables and Limit Theorems
11.1 Moment-Generating Functions 300
11.2 Sums of Independent Random Variables 308
11.3 Markov and Chebyshev Inequalities 314
11.4 Laws of Large Numbers 318
11.5 Central Limit Theorem 321
Review Problems 326
12 Stochastic Processes 330
12.2 More on Poisson Processes 330
12.3 Markov Chains 335
12.4 Continuous-Time Markov Chains 353
Review Problems 362
Companion for Chapter 12 367
12B Brownian Motion 367
FUNDAMENTALS
OF PROBABILITY
WITH STOCHASTIC PROCESSES
FOURTH EDITION
SAEED GHAHRAMANI
Western New England University
Springfield, Massachusetts, USA
A CHAPMAN & HALL BOOK
, C ontents
1 Axioms of Probability 1
1.2 Sample Space and Events 1
1.4 Basic Theorems 4
1.7 Random Selection of Points from Intervals 10
Review Problems 13
Companion for Chapter 1 19
1B Applications of Probability to Genetics 19
2 Combinatorial Methods 21
2.2 Counting Principle 21
2.3 Permutations 24
2.4 Combinations 27
2.5 Stirling’ Formula 42
Review Problems 42
3 Conditional Probability and Independence 47
3.1 Conditional Probability 47
3.2 The Multiplication Rule 52
3.3 Law of Total Probability 55
3.4 Bayes’ Formula 60
3.5 Independence 66
Review Problems 76
Companion for Chapter 3 80
3B More on Applications of Probability to Genetics 80
, Contents iii
4
Distribution Functions and
Discrete Random Variables
85
4.2 Distribution Functions 85
4.3 Discrete Random Variables 89
4.4 Expectations of Discrete Random Variables 94
4.5 Variances and Moments of Discrete Random Variables 100
4.6 Standardized Random Variables 107
Review Problems 107
5 Special Discrete Distributions 110
5.1 Bernoulli and Binomial Random Variables 110
5.2 Poisson Random Variable 117
5.3 Other Discrete Random Variables 125
Review Problems 132
6 Continuous Random Variables 137
6.1 Probability Density Functions 137
6.2 Density Function of a Function of a Random Variable 141
6.3 Expectations and Variances 144
Review Problems 151
7 Special Continuous Distributions 153
7.1 Uniform Random Variable 153
7.2 Normal Random Variable 158
7.3 Exponential Random Variables 166
7.4 Gamma Distribution 171
7.5 Beta Distribution 175
7.6 Survival Analysis and Hazard Function 180
Review Problems 183
8 Bivariate Distributions 187
8.1 Joint Distribution of Two Random Variables 187
8.2 Independent Random Variables 200
8.3 Conditional Distributions 209
8.4 Transformations of Two Random Variables 218
Review Problems 230
, Contents iv
9 Multivariate Distributions 238
9.1 Joint Distribution of n > 2 Random Variables 238
9.2 Order Statistics 248
9.3 Multinomial Distributions 253
Review Problems 255
10 More Expectations and Variances 260
10.1 Expected Values of Sums of Random Variables 260
10.2 Covariance 265
10.3 Correlation 274
10.4 Conditioning on Random Variables 276
10.5 Bivariate Normal Distribution 289
Review Problems 292
Companion for Chapter 10 299
10B Pattern Appearance 299
11
Sums of Independent Random 300
Variables and Limit Theorems
11.1 Moment-Generating Functions 300
11.2 Sums of Independent Random Variables 308
11.3 Markov and Chebyshev Inequalities 314
11.4 Laws of Large Numbers 318
11.5 Central Limit Theorem 321
Review Problems 326
12 Stochastic Processes 330
12.2 More on Poisson Processes 330
12.3 Markov Chains 335
12.4 Continuous-Time Markov Chains 353
Review Problems 362
Companion for Chapter 12 367
12B Brownian Motion 367
