Chapter 6: Continuous Random Variables

Learning Objectives

• Define the probability density function and cumulative distribution function and prove their fundamental relationship
• Compute expectations and variances using LOTUS and the tail integration formula
• Derive and apply the Uniform, Exponential, Gaussian, Gamma, Beta, and Student-t distributions
• Master the Q-function, its bounds (Chernoff, Mills ratio), and Craig's representation
• Apply the CDF method and change-of-variables formula to derive distributions of transformed random variables
• Handle mixed distributions using the CDF as a unifying object and the Dirac delta formalism