Solution Manual
Fundamentals Of Probability
By Saeed Ghahramani
5th Edition
,C ontents
► 1 Axioms Of Probability 1
1.2 Sample Space And Events 1
1.4 Basic Theorems 2
1.7 Random Selection Of Points From Intervals 7
Review Problems 9
► 2 Combinatorial Methods 13
2.2 Counting Principle 13
2.3 Permutations 16
2.4 Combinations 18
2.5 Stirling’ Formula 31
Review Problems 31
► 3 Conditional Probability And Independence 35
3.1 Conditional Probability 35
3.2 Law Of Multiplication 39
3.3 Law Of Total Probability 41
3.4 Bayes’ Formula 46
3.5 Independence 48
3.6 Applications Of Probability To Genetics 56
Review Problems 59
► 4 Distribution Functions And
Discrete Random Variables 63
4.2 Distribution Functions 63
4.3 Discrete Random Variables 66
4.4 Expectations Of Discrete Random Variables 71
4.5 Variances And Moments Of Discrete Random Variables 77
4.6 Standardized Random Variables 83
Review Problems 83
,Iv Contents
► 5 Special Discrete Distributions 87
5.1 Bernoulli And Binomial Random Variables 87
5.2 Poisson Random Variable 94
5.3 Other Discrete Random Variables 99
Review Problems 106
► 6 Continuous Random Variables 111
6.1 Probability Density Functions 111
6.2 Density Function Of A Function Of A Random Variable 113
6.3 Expectations And Variances 116
Review Problems 123
► 7 Special Continuous Distributions 126
7.1 Uniform Random Variable 126
7.2 Normal Random Variable 131
7.3 Exponential Random Variables 139
7.4 Gamma Distribution 144
7.5 Beta Distribution 147
7.6 Survival Analysis And Hazard Function 152
Review Problems 153
► 8 Bivariate Distributions 157
8.1 Joint Distribution Of Two Random Variables 157
8.2 Independent Random Variables 166
8.3 Conditional Distributions 174
8.4 Transformations Of Two Random Variables 183
Review Problems 191
► 9 Multivariate Distributions 200
9.1 Joint Distribution Of N > 2 Random Variables 200
9.2 Order Statistics 210
9.3 Multinomial Distributions 215
Review Problems 218
, Contents
►
V
10 More Expectations And Variances
222
10.1 Expected Values Of Sums Of Random Variables 222
10.2 Covariance 227
10.3 Correlation 237
10.4 Conditioning On Random Variables 239
10.5 Bivariate Normal Distribution 251
Review Problems 254
► 11 Sums Of Independent Random
Variables And Limit Theorems 261
11.1 Moment-Generating Functions 261
11.2 Sums Of Independent Random Variables 269
11.3 Markov And Chebyshev Inequalities 274
11.4 Laws Of Large Numbers 278
11.5 Central Limit Theorem 282
Review Problems 287
► 12 Stochastic Processes 291
12.2 More On Poisson Processes 291
12.3 Markov Chains 296
12.4 Continuous-Time Markov Chains 315
12.5 Brownian Motion 326
Review Problems 331
Fundamentals Of Probability
By Saeed Ghahramani
5th Edition
,C ontents
► 1 Axioms Of Probability 1
1.2 Sample Space And Events 1
1.4 Basic Theorems 2
1.7 Random Selection Of Points From Intervals 7
Review Problems 9
► 2 Combinatorial Methods 13
2.2 Counting Principle 13
2.3 Permutations 16
2.4 Combinations 18
2.5 Stirling’ Formula 31
Review Problems 31
► 3 Conditional Probability And Independence 35
3.1 Conditional Probability 35
3.2 Law Of Multiplication 39
3.3 Law Of Total Probability 41
3.4 Bayes’ Formula 46
3.5 Independence 48
3.6 Applications Of Probability To Genetics 56
Review Problems 59
► 4 Distribution Functions And
Discrete Random Variables 63
4.2 Distribution Functions 63
4.3 Discrete Random Variables 66
4.4 Expectations Of Discrete Random Variables 71
4.5 Variances And Moments Of Discrete Random Variables 77
4.6 Standardized Random Variables 83
Review Problems 83
,Iv Contents
► 5 Special Discrete Distributions 87
5.1 Bernoulli And Binomial Random Variables 87
5.2 Poisson Random Variable 94
5.3 Other Discrete Random Variables 99
Review Problems 106
► 6 Continuous Random Variables 111
6.1 Probability Density Functions 111
6.2 Density Function Of A Function Of A Random Variable 113
6.3 Expectations And Variances 116
Review Problems 123
► 7 Special Continuous Distributions 126
7.1 Uniform Random Variable 126
7.2 Normal Random Variable 131
7.3 Exponential Random Variables 139
7.4 Gamma Distribution 144
7.5 Beta Distribution 147
7.6 Survival Analysis And Hazard Function 152
Review Problems 153
► 8 Bivariate Distributions 157
8.1 Joint Distribution Of Two Random Variables 157
8.2 Independent Random Variables 166
8.3 Conditional Distributions 174
8.4 Transformations Of Two Random Variables 183
Review Problems 191
► 9 Multivariate Distributions 200
9.1 Joint Distribution Of N > 2 Random Variables 200
9.2 Order Statistics 210
9.3 Multinomial Distributions 215
Review Problems 218
, Contents
►
V
10 More Expectations And Variances
222
10.1 Expected Values Of Sums Of Random Variables 222
10.2 Covariance 227
10.3 Correlation 237
10.4 Conditioning On Random Variables 239
10.5 Bivariate Normal Distribution 251
Review Problems 254
► 11 Sums Of Independent Random
Variables And Limit Theorems 261
11.1 Moment-Generating Functions 261
11.2 Sums Of Independent Random Variables 269
11.3 Markov And Chebyshev Inequalities 274
11.4 Laws Of Large Numbers 278
11.5 Central Limit Theorem 282
Review Problems 287
► 12 Stochastic Processes 291
12.2 More On Poisson Processes 291
12.3 Markov Chains 296
12.4 Continuous-Time Markov Chains 315
12.5 Brownian Motion 326
Review Problems 331
