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Homework:
- Hw1 (with solutions) (assigned on Thursday January 25th, due on Thursday February 8th).
- Hw2 (with solutions) (assigned on Thursday March 1st, due on Thursday March 15th).
- Hw3 (with solutions) (assigned on Tuesday April 10th, due on Thursday April 26th). Instructions: Print out the homework, write down the solutions on it and staple it before you turn it in. Make sure that your solutions are correct, nicely written and well explained. Show you work, write each step of your reasonings.
Suggested Problems:
- Chapter 5: 4, 6 p 199, 1 p 202, 3, 7, 8 p 218/219, 2, 3, 4, 7, 9, 10, 14, 15, 16 p 230--232, 1, 5, 7 p 237/238
- Chapter 7: 1, 5 p 317/318, 3 p 322
- Chapter 8 (Markov Chains and Populations genetics): 4, 10, 15, 17
Exams:
- Exam 1 (with solutions) (Thursday, February 15th): chapter 5. One cheat sheet allowed (handwritten, maximum size US Letter, you can write on one side only)
- section 5.1: DeMorgan laws, conditional probabilities, Bayes Formula, Total probability formula, General version of Bayes' theorem, independence
- section 5.3: Notion of continuous RVs, probability density function, cumulative distribution function, expected value, variance and standard deviation, Normal distributions (pdf, E, Var), Exponential distributions (pdf, E, Var), Uniform distributions (pdf, E, Var), Central limit Theorem.
- section 5.4: Notion of discrete RVs, Probability mass function, c.d.f., expected value, variance and standard deviation, Bernouilli dist. (pf, E, Var), Binomial dist. (pf, E, Var), Poisson dist dist. (pf, E, Var), Geometric dist. (pf, E, Var), likelihod function and maximum likelihood estimates
- section 5.5: Joint probability mass/density function, joint cumulative distribution function, marginal distributions, characterization of independence.
- Exam 2 (with solutions) (Thursday, March 22nd): chapter 5, section 6, chapter 7, sections 1-4. One cheat sheet allowed (handwritten, maximum size US Letter, you can write on one side only)
- section 5.6: Covariance and its properties, dimensionless correlation coefficient and its properties
- section 7.1: Continuous and Discrete Stochastic Processes, index set, state space
- section 7.2: Randomizing discrete dynamics, Environmental and demographic stochasticity
- section 7.3: Random walk, difference equations, boundary conditions
- section 7.4: Linear differential equations, Simple Birth Process, Kolmogorov equations, interevent times
Final:
- Thursday May 10th 9:45pm-11:45pm No calculator. One cheat sheet allowed (handwritten, maximum size US Letter, you can write on both sides)
- Program:
- Markov Chains (probability transition matrices, absorption probabilities, stationary distributions, convergence to stationary distributions,...)
- Wiener processes, Stochastic Differential equations, Ito SDEs, SDEs from Markov models
- Simple Random Walk
- Population Dynamics and Kolmogorov differential equations
- Discrete and Continuous Random variables: classic distributions (Bernouilli, Binomial, Geometric, Poisson, Normal, Exponential,...), Cumulative Distribution Function, Expected value, Variance
- Joint Probability distributions
- Covariance, Dimensionless Correlation coefficient
- Independence
- Conditional probabilities, Bayes' Theorem
- Central limit theorem
- Difference equations
- Maximum Likelihood Estimator.
Schedule
| Date | Section Covered | Problems Solved | Reading assignments |
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| 9 January | Beginning Section 5.1: Concepts of Probability, introductory examples and definitions | | |
| 11 January | End Section 5.1: Concepts of Probability, introductory examples and definitions | 2, 7 p 198/199 | Examples 5.1, 5.4, 5.6, 5.7 (from section 5.1) |
| 16 January | Section 5.2: the Hardy-Weinberg law | 8 p 198/199 | |
| 18 January | Section 5.3.1: Normal Distributions | 2 p 202, 1 p 217 | |
| 23 January | Section 5.3.2: General Continuous Random Variables | | |
| 25 January | End Section 5.3.2: General Continuous Random Variables | 4 p 218 | |
| 30 January | End Section 5.3.3: Random search, hazard rates, section Section 5.3.4: Central limit theorem | 13 p 220 | |
| 1 February | Section 5.4: Discrete Random Variables | 1, 5 p 231 | |
| 6 February | Section 5.4: Discrete Random Variables | 8 p 231 | Examples 5.29 (Neg. Binomial) p 223, 5.31 (Multi.), 5.32 (Hypergeom.) p 225 |
| 8 February | Section 5.5: Joint probability distributions, joint cumulative distribution functions. marginal density functions, | 17 p 232, Beginning 3 p 237 | |
| 13 February | Section 5.5: expected value of g(X,Y), characterization of independence of RV. Review Exam 1 | End 3 p 237, 4 p 237 | |
| 20 February | Section 5.6: Covariance, properties of the covariance | | |
| 22 February | Section 5.6: Dimensionless correlation coefficient and its properties, covariance matrix | 2, 3 p 243 | |
| 27 February | Section 7.1: Intro to stochastic Processes, Beginning section 7.2: Randomizing Discrete Dynamics | | Example 7.2, Figure 7.3 section 7.1 |
| 1 March | Section 7.2: Randomizing Discrete Dynamics, Demographic and Environmental Stochasticity | 3 p 318 | |
| 6 March | End Section 7.2: Demographic and Environmental Stochasticity, Beginning Section 7.3: Random Walk | 6 p 318 | Environmental Stochasticity p 315/316 |
| 8 March | Section 7.3: Random Walk, Beginning Section 7.4: Linear Differential Equations | 1, 2 p 325 | |
| 13 March | Section 7.4: Simple birth process | | |
| 15 March | End Section 7.4: Simple birth process, inter event times. Markov chains | | |
| 20 March | Markov chains, Random mating, absorbing states | | Pages 158-160, chapter on Markov Chains |
| 3 April | Absorption probabilities, fundamental matrix of a homogeneous Markov Chain, application to simple Random Walk | | Proof Thm. 8.5 p 164, Examples 1 and 3 p 164 and 166, chapter on Markov Chains |
| 5 April | Mean time to Absorption, stationary distributions | | Proof Thm. 8.6 p 168 |
| 10 April | Regular Markov Chains, Application to gene mutations | |
| 12 April | Practice Problems on Markov Chains | 5, 6, 18 from chapter 8 | |
| 17 April | Sections 7.5.1, 7.5.2: Brownian Motion, Diffusion Equation, Wiener Processes | 1 p 337 | |
| 19 April | Section 7.5.2: Wiener Processes | 2, 3 p 337 | |
| 24 April | Section 7.5.3: Stochastic Differential Equations | 1 p 341 | |
| 26 April | Section 7.6: SDEs from Markov Models | 2 p 345 | |
Lab (Math 305L):
- Practice problems and Homework, Lab 1 (assigned on Thursday January 11th, due on Thursday January 18st).
- Practice problems and Homework, Lab 2 (assigned on Monday January 22nd, due on Monday January 29th).
- Practice problems and Homework, Lab 3 (assigned on Tuesday January 30th, due on Tuesday February 6th).
- Practice problems and Homework, Lab 4 (assigned on Thursday February 8th, due on Thursday February 15th).
- Practice problems and Homework, Lab 5 (assigned on Tuesday February 20th, due on Tuesday February 27th).
- Practice problems and Homework, Lab 6 (assigned on Tuesday February 27th, due on Tuesday March 6th).
- Practice problems and Homework, Lab 7, labwk7.pdf (assigned on Tuesday March 13th, due on Tuesday March 20th).
- Practice problems and Homework, Lab 8 (assigned on Tuesday April 3rd, due on Tuesday April 10th).
- Practice problems and Homework, Lab 9 (assigned on Thursday April 18th, due on Thursday May 3rd).
Projects (Math 305L):
- Instructions: choose a model to study from one of the published research papers below. Each Student must choose a different paper (let me know your choice by email, first come, first served). You must simulate the model using Matlab (or Octave), and reproduce the results from the paper. In addition, you must formulate a question that is not fully addressed in the paper, and modify the model to study it, or extend the analysis of the behavior of the model beyond that of the paper. You must type a report that includes the model you studied, the reproduction of the results from the paper, the code you wrote, the question you formulated, your analysis and results. The report is due by Wednesday, May 2nd.
- List of articles:
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