2 - Random Variables

Summary of notes from lectures by Sheldon Jacobson for the course IE 410 at UIUC in Fall 2021.

• Definitions: random variable, CDF.
• Distributions: Bernouilli, Binomial, Geometric, Negative binomial, Poisson.
• Approximating Binomial using Poisson, Poisson as limiting case of binomial.
• Distributions: Exponential, Gamma, Normal.
• Theorem: X ~ N(μ,σ) implies aX+b ~ N(aμ+b,aσ).
• Buffon Needle problem.
• Definition: expectation and variance.
• News vendor problem.
• Definition: Joint CDF and PMF.
• If X and Y are independent, then E(XY) = E(X)E(Y).
• Definition: covariance.
• Sum of Poisson randvars.
• Sum of exponential randvars.
• Theorem: Joint distribution of Y_1 and Y_2, where Y_1 = f_1(X_1, X_2) and Y_2 = f_2(X_1, X_2).
• Coupon collector problem.
• Definition: Moment-generating function (MGF).
• Theorem: MGF uniquely defines probability distribution.
• Theorem: MGF of sum of indep randvars is product of MGFs of those randvars.
• Markov's inequality.
• Chebychev's inequality.
• One-sided chebychev's inequality.
• Jensen's inequality.
• Weak and strong law of large numbers.
• Central limit theorem.