Probability and Statistics: Variance and Covariance Calculations

This lecture covers fundamental concepts in probability and statistics, focusing on variance and covariance calculations. The instructor begins by discussing the importance of feedback questionnaires and then introduces the formulas of Koenig for calculating variance and covariance between random variables. The lecture includes a detailed proof of the variance formula, emphasizing the relationship between the expectation of a variable and its square. The instructor explains how to compute the density of sums of continuous random variables, particularly Gaussian and gamma distributions, and highlights the stability of the Gaussian distribution. The lecture also addresses transformations of random variables, including affine transformations, and their impact on distributions. The instructor illustrates how to derive the density functions for transformed variables and discusses mixtures of random variables, providing examples related to real-world applications. The session concludes with a discussion on the moments of random variables and their significance in understanding distributions.