|Course Archives Theoretical Statistics and Mathematics Unit|
Course: Probability Theory I
Time: Currently not offered
i) Random experiments, outcomes, sample space, events. Discrete sample spaces and probability models . Equally likely setup (including examples such as Maxwell-Boltzmann, Bose-Einstein, Fermi-Dirac statistics) and combinatorial probability (examples such as Gibbs distributions (definition)). Combination of events: inclusion/exclusion, Booles inequality and Bonferronis inequality.
ii) Conditional probability: independence, law of total probability and Bayes theorem, Composite experiments: Polyas urn scheme.
iii) Discrete random variables. Standard discrete distributions(degenerate, Bernoulli, Binomial, discrete uniform, Hypergeometric, Poisson, Geometric, negative binomial). Convergence of Binomial to Poisson distribution and normal distribution (latter only statement and sketch of proof).
iv) Continuous random variables [with densities continuous except at finitely many points]; Examples of uniform, exponential, beta, gamma, normal, Cauchy, Pareto and other densities . Introduction to cumulative distribution functions (CDF) and properties. Distributions with densities. Standard univariate densities (uniform, exponential, beta, gamma, normal)
v) functions of random variables , Expectation/mean , moments, variance , computations involving indicator random variables
vi) Joint distributions of discrete random variables (multionomial distributions), linearity and monotonicity of expectations,independence, covariance, variance of a sum,computations involving indicator random variables, distribution of sum of two independent random variables. Conditional distributions, conditional expectation.
Note: In continous random variables, focus only on densities which are continous except at finitely many points and Riemann integration (proper and improper) to be introduced in an informal manner without proofs. Also, emphasize that CDF or density matters and not the underlying uncountable sample space.
(a) W. Feller: Introduction to Probability: Theory and Applications - Vol. I and II.
(b) J. Pitman: Probability.
(c) Sheldon Ross: Probability Models.
(d) Santosh S. Venkatesh: Theory of Probability - Explorations and Applications.
(e) P. G. Hoel, S. C. Port and C. J. Stones: Introduction to Probability Theory.
(f) K. L. Chung: Elementary Probability Theory with Stochastic Processes.
(g) R. Meester: A Natural Introduction to Probability Theory.
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