Part IA - Probability Based on lectures by R. Weber Notes taken by Dexter Chua 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. Basic concepts Classical probability, equally likely outcomes. Combinatorial analysis, permutations and combinations. Stirling's formula (asymptotics for log n! proved). [3 Axiomatic approach Axioms (countable case). Probability spaces. Inclusion-exclusion formula. Continuity and subadditivity of probability measures. Independence. Binomial, Poisson and geo- metric distributions. Relation between Poisson and binomial distributions. Conditional probability, Bayes's formula. Examples, including Simpson's paradox. [5] Discrete random variables Expectation. Functions of a random variable, indicator function, variance, standard deviation. Covariance, independence of random variables. Generating functions: sums of independent random variables, random sum formula, moments. Conditional expectation. Random walks: gambler's ruin, recurrence relations. Dif- ference equations and their solution. Mean time to absorption. Branching processes: generating functions and extinction probability. Combinatorial applications of generat- ing functions. [7 Continuous random variables Distributions and density functions. Expectations; expectation of a function of a random variable. Uniform, normal and exponential random variables. Memoryless property of exponential distribution. Joint distributions: transformation of random variables (including Jacobians), examples. Simulation: generating continuous random variables, independent normal random variables. Geometrical probability: Bertrand's paradox, Buffon's needle. Correlation coefficient, bivariate normal random variables. [6] Inequalities and limits Markov's inequality, Chebyshev's inequality. Weak law of large numbers. Convexity: Jensen's inequality for general random variables, AM/GM inequality. Moment generating functions and statement (no proof) of continuity theorem. State- ment of central limit theorem and sketch of proof. Examples, including sampling. [3] 1
Contents IA Probability Contents 0 Introduction 3 1 Classical probability 4 1.1 Classical probability 4 1.2 Counting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Stirling's formula 8 2 Axioms of probability 11 2.1 Axioms and definitions 11 2.2 Inequalities and formulae . . . . . . . . . . . . . . . . . . . 13 . . . 2.3 Independence 16 2.4 Important discrete distributions . 17 2.5 Conditional probability . . . . . . . . . . . . . . . . . . . . . . . 18 3 Discrete random variables 22 3.1 Discrete random variables 22 3.2 Inequalities 31 3.3 Weak law of large numbers 33 3.4 Multiple random variables 34 3.5 Probability generating functions . . . . . . . . . . . . . . . . . . 37 4 Interesting problems 43 4.1 Branching processes 43 4.2 Random walk and gambler's ruin 46 5 Continuous random variables 50 5.1 Continuous random variables . . . . . . . . . . . . . . . . . 50 . 5.2 Stochastic ordering and inspection paradox