Probability

Probability

PROBABILITY
Basic concepts

  • Classical probability, equally likely outcomes. Combinatorial analysis, permutations and combinations. Stirling’s formula (asymptotics for logn! proved). [3]

Axiomatic approach

  • Axioms (countable case). Probability spaces. Inclusion-exclusion formula. Continuity and subadditivity of probability measures. Independence. Binomial, Poisson and geometric 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. Difference equations and their solution. Mean time to absorption. Branching processes: generating functions and extinction probability. Combinatorial applications of generating 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, Box–Muller transform, rejection sampling. 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. Statement of central limit theorem and sketch of proof. Examples, including sampling. [3]

Appropriate books

  • W. Feller An Introduction to Probability Theory and its Applications, Vol. I. Wiley 1968(pdf)
  • † G. Grimmett and D. Welsh Probability: An Introduction. Oxford University Press 2nd Edition 2014(pdf)
  • † S. Ross A First Course in Probability. Prentice Hall 2009(pdf)
  • D.R. Stirzaker Elementary Probability. Cambridge University Press 1994/2004(pdf)