Chapter 5: Discrete Random Variables

Learning Objectives

• Define a random variable as a measurable function from the sample space to the reals, and distinguish discrete from continuous RVs
• Compute the PMF and CDF for discrete random variables and verify their properties
• Compute expectations via the definition and via LOTUS, and exploit linearity of expectation
• Derive and apply the variance identity $\text{Var}(X) = \mathbb{E}[X^2] - (\mathbb{E}[X])^2$
• Characterize the Bernoulli, binomial, geometric, negative binomial, Poisson, discrete uniform, and hypergeometric distributions by their PMF, mean, variance, and MGF
• Define Shannon entropy for a discrete random variable and interpret it as average surprise
• Recognize the Poisson distribution as a limit of the binomial and as the foundation for queueing theory