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Fundamentals of Probability with stochastic processes-Solutions

# Fundamentals of Probability with stochastic processes-Solutions

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Instructor's Solutions Manual Third Edition Fundamentals of ProbabilitY With Stochastic Processes SAEED GHAHRAMANI Western New England College Upper Saddle River, New Jersey 07458

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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

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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

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Chapter 1 A xioms of P robability 1.2 SAMPLE SPACE AND EVENTS 1. For 1 i, j 3, by (i, j) we mean that Vann’s card number is i , and Paul’s card number is j . Clearly, A = ( 1 , 2 ), ( 1 , 3 ), ( 2 , 3 ) and B = ( 2 , 1 ), ( 3 , 1 ), ( 3 , 2 ) . (a) Since A B = ∅ , the events A and B are mutually exclusive.

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