## Manjunath Krishnapur

This is the outline of a first course in (discrete) Probability that I gave in the summer of 2009. We used as text, the first volume of Feller's `An introduction to Probability and examples'. There are innumerable books (eg., Sheldon Ross' introductory book is quite popular and good) that contain the same concepts are available. Feller's book has the advantage of containing a spectacular array of examples and perhaps the disadvantage(?) of not being linear. Solving problems was one of the main activities in the course and is essential to comprehend the material. Apart from the problems in the book, I gave out a few problem sets.

### Contents

• Probability space. (Countable sample spaces) Chapter 1 [Feller vol. 1]
• Inclusion-exclusion principle. Occupancy and Matching problems with Poisson asymptotics. Chapter 4
• Conditional Probability and Independence. Chapter 5.1, 5.3
• Polya's urn scheme - exchangeability. Chapter 5.2
• Random variables and their distributions. Chapter 9.1
• Bernoulli, Binomial, Geometric, Poisson, Hypergeometric distributions. Parts of chapter 6
• Expectation of a random variable. Variance and Moments. Chapter 9.2-9.4
• Joint distribution and independence of random variables. Conditional distributions. Chapter 9.1, 9.5
• Markov's and Chebyshev's inequalities. Weak law under second moment assumption. Chapter 9.6, 10.1, 10.2
• CLT for sums of Bernoullis. Poisson limit for rare events. Chapter 7.2
• Simple random walk in one dimension. Ballot problem. Gambler's ruin. Recurrence. Parts of Chapter 3 and 14
• Simulating random variables on a computer. Not in Feller's book!
• Continuous distributions. An introduction. From volume 2 or elsewhere.
• Uniform, Normal. Exponential, Gamma, Beta distributions. From volume 2 or elsewhere.