Vectors of Random Variables: Empirical Distributions

This lecture covers the concepts of vectors of random variables and empirical distributions. The instructor begins by discussing the definition of a vector of random variables, emphasizing that each component is itself a random variable. The lecture explains the difference between independent and dependent random variables, and how to characterize their distributions through probability mass functions and cumulative distribution functions. The instructor introduces the concept of empirical distributions, illustrating how to calculate empirical probabilities based on observed data, such as the heights of students in a class. The discussion includes the calculation of empirical means and variances, highlighting their significance in statistics. The instructor also addresses the importance of independence in random variables and how it relates to their distributions. The lecture concludes with an introduction to the law of large numbers, explaining how the average of a sample converges to the expected value as the sample size increases, setting the stage for further exploration in future lectures.