MTH311A - Probability Theory - I

 Sets and set operations, Sample space, Sigma fields, Measurable spaces, Events. Measure spaces, Caratheodorys extension theorem, Construction of measures, Product spaces, Product measures. Probability measurer and its properties. Independence of events. Measurable functions, Approximations through simple functions, Random variables. Induced measures and probability distribution functions: discrete, continuous and absolutely continuous, one to one correspondence with induced probability measure, decomposition. Independence of random variables, Borel Cantelli lemmas. Integration in measure spaces, Expectation, Fatous lemma, Monotone convergence and dominated convergence theorems, Uniform integrability, Markov, Chebyshev, Cauchy Schwarz, Minkowski, Holder, Jensen and Lyapunov inequalities. Absolute continuity of measures, Randon Nikodym theorem, Conditional expectation, Conditional probability measures. Fubinis theorem, Convolution. Functions of random variables, Jacobian theorem.