Expectation and Variance

This lecture delves into the concepts of expectation and variance for discrete random variables. The instructor explains how to calculate the expectation of a transformed variable, explores properties of expectation, and demonstrates the calculation of moments and variance. Key topics include linearity of expectation, properties of variance, moments of a distribution, and the relationship between variance and moments. The lecture also covers the importance of independence in calculating expectations and introduces the concept of moments as analogs to physical properties like center of gravity and moment of inertia. Through various examples and proofs, the instructor illustrates the practical applications of these fundamental probabilistic concepts.