Distribution of Random Variables

This lecture introduces the concept of the distribution of a random variable, starting with an abstract definition using probability measures. It then delves into the cumulative distribution function, which provides a more intuitive way to describe the distribution. The properties of a CDF, including being non-decreasing and right-continuous, are discussed. The lecture further explores specific cases of random variables, focusing on discrete random variables with probability mass functions and continuous random variables with probability density functions. The relationship between PDFs and CDFs is explained, highlighting the continuous nature of CDFs. The lecture concludes by debunking the misconception that all CDFs can be decomposed into discrete and continuous parts.