# Part IA - Probability

## Lectured by R. Weber, Lent 2015

These notes are not endorsed by the lecturers, and I have modified them (often significantly) after lectures. They are nowhere near accurate representations of what was actually lectured, and in particular, all errors are almost surely mine.

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# Contents

- V Full version
- 0 Introduction
- 1 Classical probability
- 2 Axioms of probability
- 2.1 Axioms and definitions
- 2.2 Inequalities and formulae
- 2.3 Independence
- 2.4 Important discrete distributions
- 2.5 Conditional probability
- 3 Discrete random variables
- 3.1 Discrete random variables
- 3.2 Inequalities
- 3.3 Weak law of large numbers
- 3.4 Multiple random variables
- 3.5 Probability generating functions
- 4 Interesting problems
- 5 Continuous random variables
- 5.1 Continuous random variables
- 5.2 Stochastic ordering and inspection paradox
- 5.3 Jointly distributed random variables
- 5.4 Geometric probability
- 5.5 The normal distribution
- 5.6 Transformation of random variables
- 5.7 Moment generating functions
- 6 More distributions
- 6.1 Cauchy distribution
- 6.2 Gamma distribution
- 6.3 Beta distribution*
- 6.4 More on the normal distribution
- 6.5 Multivariate normal
- 7 Central limit theorem
- 8 Summary of distributions