### Syllabus

### Homework

### Sample Tests

- Test 1
- Test 2
- Test 3
### Solution to Homework Assignments

- Homework 1
- Homework 2
- Homework 3
- Homework 4
- Homework 5
- Homework 6
- Homework 7
- Homework 8
- Homework 9
- Homework 10
- Homework 11
### Solutions to Tests

- Test 1
- Test 2
- Test 3
### Class Diary

**Wedensday, 01/17:**sample spaces and events, probabilities, conditional probabilities, independence**Read:**Sections 1.2--1.5**Friday, 01/19:**Bayes' formula, random variables, discrete random variables, (absolutely) continuous random variables**Read:**Sections 1,6, 2.1--2.3**Monday, 01/22:**expectation of a random variable, expectation of a function of a random variable, variance of a random variable**Read:**Section 2.4**Wedensday, 01/24:**joint distributions, marginal distributions, joint mass functions, marginal mass functions, joint densities, marginal densities, covariances**Read:**Section 2.5**Friday, 01/26:**distributions of sums of independenr random variables, momemnt generating functions, laws of large numbers, central limit theorems**Read:**Sections 2.5, 2.6, 2.8**Monday, 01/29:**stochastis processes, conditional mass functions, conditional density functions, condistional expectations**Read:**Section 2.9, 3.2 and 3.3**Wedensday, 01/31:**computing expectations by conditioning, computing variances by conditioning**Read:**Section 3.4**Friday, 02/02:**computing probabilities by conditioning**Read:**Section 3.5**Monday, 02/05:**compound random variable identity**Read:**Section 3.7**Wedensday, 02/07:**Markov chains, transition probabilities, n-step transition probabilities, Chapman-Kolmogorov equations**Read:**Sections 4.1 and 4.2**Friday, 02/09:**Chapman-Kolmogorov equations, n-step transition probabilities**Read:**Section 4.2**Monday, 02/12:**n-step transition probabilities (continued)**Read:**Section 4.2**Wedensday, 02/14:**classification of states**Read:**Section 4.3**Friday, 02/16:**recurrence, transience, random walks.**Read:**Section 4.3**Monday, 02/19:**long-run proportions and limiting probabilities**Read:**Section 4.4**Wedensday, 02/21:**stationary distributions. Review**Read:**Section 4.4**Friday, 02/23:**Test 1 will cover materials up to (including) Section 4.3. Test 1: score distribution (out of 80 points): 80**2**79**1**76**1**75**2**74**1**73**1**69**1**66**1**65**1**61**1**60**1**57**1**55**2****Monday, 02/26:**Gambler's ruin, mean time spent in transient states**Read:**Section 4.5.1 and Section 4.6**Wedensday, 02/28:**branching processes**Read:**Section 4.7**Friday, 03/02:**branching processes (continued), time reversible Markov chains**Read:**Sections 4.7 and 4.8**Monday, 03/05:**time reversible Markov chains (continued), exponential distributions**Read:**Sections 4.8 and 5.2**Wedensday, 03/07:**properties of exponential random variables, failure rates, hyperexponential random variables**Read:**Section 5.2**Friday, 03/09:**hypoexpential random variables, Coxian random variables, counting processes**Read:**Sections 5.2 and 5.3**Monday, 03/12:**Poisson processes, interarrival times, waiting time until the nth event**Read:**Section 5.3**Wedensday, 03/14:**further properties of Poisson processes conditional distribution of arrival times.**Read:**Section 5.3.4**Friday, 03/16:**nonhomogeneous Poisson processes, compound Poisson processes**Read:**Section 5.4.1 and 5.4.2**Moday, 03/26:**Continuous-time Markov chains, birth and death processes**Read:**Sections 6.2 and 6.3**Wedensday, 03/28:**Review for Test 2, birth and death processes**Read:**Section 6.3**Friday, 03/30:**Test 2: Materials on Test 2 include all Chapter 4, and Section 5.2 and 5.3. Score distributions: 80**2**76**1**75**3**72**1**70**2**65**2**64**1**58**1****Monday, 04/02:**Transition probabilities, Example 6.8, expected time for a birth and death process to go from a state i to a state j>i**Read:**Sections 6.3 and 6.4**Wedensday, 04/04:**variance of the time for a birth and death process to go from a state i to a state j>i, Lemmas 6.2 and 6.3**Read:**Sections 6.3 and 6.4**Friday, 04/06:**backward equations and forward equations**Read:**Section 6.4**Monday, 04/09:**limiting probabilities**Read:**Section 6.5**Wedensday, 04/11:**limiting probabilities**Read:**Section 6.5**Friday, 04/13:**structure functions, minimal paths, minimal path sets**Read:**Section 9.2**Monday, 04/16:**minimal cut, minimal cut sets, reliability function, systems with independent components**Read:**Sections 9.2 and 9.3**Wedensday, 04/18:**Bounds on the reliability function (the inclusion-exclusion method)**Read:**Section 9.4**Friday, 04/20:**the second method for bounding the reliability funcions, System life as a function of component lives**Read:**Sections 9.4 and 9.5**Monday, 04/23:**System life as a function of component lives; expected system lifetime**Read:**Sections 9.5 and 9.6**Wedensday, 04/25:**expected system lifetime**Read:**Section 9.6**Friday, 04/27:**Test 3: Materials on Test 3 include Sections 5.3, 5.4, 6.2, 6.3, 6.4, 6.5, 9.2, 9.3, Test 3 score distribution (out of 80 points): 80**4**78**1**77**1**76**1**75**1**73**1**71**1**65**1**40**1****Monday, 04/30:****Read:**Section**Wedensday, 05/02:**