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E-grāmata: Probability and Simulation

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Probability and SimulationPalielināt
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This undergraduate textbook presents an inquiry-based learning course in stochastic models and computing designed to serve as a first course in probability. Its modular structure complements a traditional lecture format, introducing new topics chapter by chapter with accompanying projects for group collaboration. The text addresses probability axioms leading to Bayes theorem, discrete and continuous random variables, Markov chains, and Brownian motion, as well as applications including randomized algorithms, randomized surveys, Benfords law, and Monte Carlo methods.

Adopting a unique application-driven approach to better study probability in action, the book emphasizes data, simulation, and games to strengthen reader insight and intuition while proving theorems. Additionally, the text incorporates codes and exercises in the Julia programming language to further promote a hands-on focus in modelling. Students should have prior knowledge of single variable calculus.

Giray Ökten received his PhD from Claremont Graduate University. He has held academic positions at University of Alaska Fairbanks, Ball State University, and Florida State University. He received a Fulbright U.S. Scholar award in 2015. He is the author of an open access textbook in numerical analysis, First Semester in Numerical Analysis with Julia, published by Florida State University Libraries, and a co-author of a childrens math book, The Mathematical Investigations of Dr. O and Arya, published by Tumblehome. His research interests include Monte Carlo methods and computational finance.





 
1 Probability
1(20)
1.1 Axioms of probability
1(2)
1.2 Verifying polynomial identities
3(3)
1.2.1 Randomized algorithm
3(3)
1.3 Project 1: Verifying identities using Julia
6(1)
1.4 Verifying matrix multiplication: Freivalds' algorithm
7(2)
1.5 Project 2: Analysis of Freivalds' algorithm
9(2)
1.6 Conditional probability and randomized surveys
11(4)
1.6.1 Warner model
12(1)
1.6.2 Simmons Model
13(2)
1.7 Project 3: A survey with three choices
15(1)
1.8 Bayes' theorem
16(3)
1.9 Project 4: The Haunting of Hill House
19(2)
2 Discrete Random Variables
21(26)
2.0.1 Expectation of a function of a random variable
23(2)
2.1 Discrete uniform random variables
25(3)
2.2 Project 5: Benford's law
28(4)
2.3 Bernoulli, binomial, geometric, Poisson random variables
32(5)
2.3.1 Bernoulli and binomial random variables
32(1)
2.3.2 Geometric random variables
33(2)
2.3.3 Poisson random variables
35(2)
2.4 Project 6: Resurrect the Beetle!
37(2)
2.5 Conditional expectation
39(7)
2.5.1 Computing probabilities by conditioning
41(1)
2.5.2 Martingales
42(4)
2.6 Project 7: A professor's trick
46(1)
3 Continuous Random Variables
47(34)
3.1 Uniform random variables
52(3)
3.1.1 Strong law of large numbers
52(3)
3.2 Project 8: Monte Carlo integration
55(3)
3.3 Exponential and normal random variables
58(14)
3.3.1 The Normal random variable
64(8)
3.4 Project 9: Florida Panther
72(3)
3.5 k-distribution and #2-test
75(4)
3.6 Project 10: Can humans generate random numbers?
79(2)
4 Markov Chains
81(18)
4.1 Introduction to Markov chains
81(5)
4.1.1 Higher transition probabilities
83(3)
4.2 Project 11: Analyzing a die game with Markov chains
86(1)
4.3 State vectors and limiting probabilities
87(10)
4.3.1 Limiting behavior of the state vector
88(4)
4.3.2 Parrando's paradox
92(5)
4.4 Project 12: Market share of shoe brands
97(2)
5 Brownian Motion
99(16)
5.1 Brownian motion
99(5)
5.1.1 Simulating Brownian motion
101(3)
5.2 Project 13: Modeling insect movement
104(2)
5.3 Geometric Brownian motion
106(6)
5.3.1 Simulating stock prices
108(4)
5.4 Project 14: Option pricing
112(3)
Appendix A Benford's law 115(8)
Appendix B Data for Project 12: Market share of shoe brands 123(2)
Solutions 125(24)
References 149(2)
Index 151
Giray Ökten received his PhD from Claremont Graduate University. He has held academic positions at University of Alaska Fairbanks, Ball State University, and Florida State University. He received a Fulbright U.S. Scholar award in 2015. He is the author of an open access textbook in numerical analysis, First Semester in Numerical Analysis with Julia, published by Florida State University Libraries, and a co-author of a childrens math book, The Mathematical Investigations of Dr. O and Arya, published by Tumblehome. His research interests include Monte Carlo methods and computational finance.
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