Instructor's Solutions Manual - Fundamentals Of Probability with....pdf

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I n s t r u c t o r ' s S o l u t i o n s M a n u a l Third Edition F u n d a m e n t a l s o f P r o b a b i l i t Y With Stochastic Processes SAEED GHAHRAMANI Western New England College Upper Saddle River, New Jersey 07458

C o n t e n t s 1 Axioms of Probability 1.2 Sample Space and Events 1.4 Basic Theorems 1.7 Random Selection of Points from Intervals 2 1 7 Review Problems 9 2 Combinatorial Methods 2.2 Counting Principle 2.3 Permutations 2.4 Combinations 2.5 Stirling’ Formula Review Problems 16 18 13 31 31 3 Conditional Probability and Independence 35 3.1 Conditional Probability 3.2 Law of Multiplication 3.3 Law of Total Probability 3.4 Bayes’ Formula 46 3.5 3.6 Applications of Probability to Genetics Independence 41 39 48 56 Review Problems 59 4 Distribution Functions and Discrete Random Variables 63 4.2 Distribution Functions 4.3 Discrete Random Variables 4.4 Expectations of Discrete Random Variables 4.5 Variances and Moments of Discrete Random Variables 4.6 Standardized Random Variables 66 71 83 Review Problems 83 1 13 35 63 77

iv Contents 5 Special Discrete Distributions 5.1 Bernoulli and Binomial Random Variables 5.2 Poisson Random Variable 94 5.3 Other Discrete Random Variables 99 87 Review Problems 106 6 Continuous Random Variables 6.1 Probability Density Functions 6.2 Density Function of a Function of a Random Variable 6.3 Expectations and Variances 123 Review Problems 111 116 7 Special Continuous Distributions 126 131 7.1 Uniform Random Variable 7.2 Normal Random Variable 7.3 Exponential Random Variables 7.4 Gamma Distribution 144 7.5 Beta Distribution 7.6 Survival Analysis and Hazard Function 147 139 Review Problems 153 8 Bivariate Distributions 152 Joint Distribution of Two Random Variables Independent Random Variables 8.1 8.2 8.3 Conditional Distributions 8.4 Transformations of Two Random Variables 166 174 Review Problems 191 9 Multivariate Distributions Joint Distribution of n > 2 Random Variables 9.1 9.2 Order Statistics 210 9.3 Multinomial Distributions 218 Review Problems 215 157 183 200 113 87 111 126 157 200

10 More Expectations and Variances Contents v 222 10.1 Expected Values of Sums of Random Variables 10.2 Covariance 10.3 Correlation 10.4 Conditioning on Random Variables 10.5 Bivariate Normal Distribution 227 237 239 251 222 Review Problems 254 11 Sums of Independent Random Variables and Limit Theorems 11.1 Moment-Generating Functions 261 11.2 Sums of Independent Random Variables 11.3 Markov and Chebyshev Inequalities 11.4 Laws of Large Numbers 11.5 Central Limit Theorem 278 282 Review Problems 287 269 274 12 Stochastic Processes 261 291 12.2 More on Poisson Processes 12.3 Markov Chains 12.4 Continuous-Time Markov Chains 12.5 Brownian Motion Review Problems 326 331 296 291 315

Chapter 1 A x i om s o f Pr o b a b i l i t y 1.2 SAMPLE SPACE AND EVENTS j. Clearly, A = 1. For 1 ≤ i, j ≤ 3, by (i, j ) we mean that Vann’s card number is i, and Paul’s card number is (1, 2), (1, 3), (2, 3) (2, 1), (3, 1), (3, 2) (a) Since A ∩ B = ∅, the events A and B are mutually exclusive. (b) None of (1, 1), (2, 2), (3, 3) belongs to A∪ B. Hence A∪ B not being the sample space shows that A and B are not complements of one another. . and B = 2. S = {RRR, RRB, RBR, RBB, BRR, BRB, BBR, BBB}. 3. {x: 0 < x < 20}; {1, 2, 3, . . . , 19}. 4. Denote the dictionaries by d1, d2; the third book by a. The answers are {d1d2a, d1ad2, d2d1a, d2ad1, ad1d2, ad2d1} and {d1d2a, ad1d2}. 5. EF : One 1 and one even. EcF : One 1 and one odd. EcF c: Both even or both belong to {3, 5}. 6. S = {QQ, QN, QP , QD, DN, DP , N P , N N, P P}. (a) {QP}; (b) {DN, DP , N N}; (c) ∅. ∪ ∪ ≤ x ≤ 8 1 4 4 x: 8 3 4 ≤ x ≤ 9 1 6 . 7. S = x: 7 ≤ x ≤ 9 1 x: 7 3 8. E ∪ F ∪ G = G: If E or F occurs, then G occurs. x: 7 ≤ x ≤ 7 1 ; 6 4 EF G = G: If G occurs, then E and F occur. 9. For 1 ≤ i ≤ 3, 1 ≤ j ≤ 3, by aibj we mean passenger a gets off at hotel i and passenger b gets off at hotel j. The answers are {aibj : 1 ≤ i ≤ 3, 1 ≤ j ≤ 3} and {a1b1, a2b2, a3b3}, respectively. 10. (a) (E ∪ F )(F ∪ G) = (F ∪ E)(F ∪ G) = F ∪ EG.

2 Chapter 1 Axioms of Probability (b) Using part (a), we have (E ∪ F )(Ec ∪ F )(E ∪ F c) = (F ∪ EEc)(E ∪ F c) = F (E ∪ F c) = F E ∪ F F c = F E. 11. (a) AB cCc; (b) A ∪ B ∪ C; (e) AB cCc ∪ AcB cC ∪ AcBCc; (d) ABCc ∪ AB cC ∪ AcBC; (c) AcB cCc; (f) (A − B) ∪ (B − A) = (A ∪ B) − AB. 12. If B = ∅, the relation is obvious. If the relation is true for every event A, then it is true for S, the sample space, as well. Thus S = (B ∩ Sc) ∪ (B c ∩ S) = ∅ ∪ B c = B c, 13. Parts (a) and (d) are obviously true; part (c) is true by DeMorgan’s law; part (b) is false: throw showing that B = ∅. ∞ a four-sided die; let F = {1, 2, 3}, G = {2, 3, 4}, E = {1, 4}. 14. (a) n=1 An; (b) 37 n=1 An. 15. Straightforward. 16. Straightforward. 17. Straightforward. 18. Let a1, a2, and a3 be the ﬁrst, the second, and the third volumes of the dictionary. Let a4, a5, a6, and a7 be the remaining books. Let A = {a1, a2, . . . , a7}; the answers are x1x2x3x4x5x6x7: xi ∈ A, 1 ≤ i ≤ 7, and xi = xj if i = j S = and x1x2x3x4x5x6x7 ∈ S: xixi+1xi+2 = a1a2a3 for some i, 1 ≤ i ≤ 5 , respectively. ∞ ∞ 20. Let B1 = A1, B2 = A2 − A1, B3 = A3 − (A1 ∪ A2), . . . , Bn = An − n=m An. m=1 19. n−1 i=1 Ai, . . . . 1.4 BASIC THEOREMS 1. No; P (sum 11) = 2/36 while P (sum 12) = 1/36. 2. 0.33 + 0.07 = 0.40.

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