Introduction to Continuous Random Variables: Probability Distributions

This lecture provides an introduction to continuous random variables and their associated probability distributions. It begins with an overview of probability density functions and cumulative distribution functions, highlighting their significance in statistical analysis. The instructor explains various continuous distributions, including uniform, normal, exponential, and gamma distributions, detailing their characteristics and applications. The lecture emphasizes the importance of the Central Limit Theorem, which states that the sample mean of a sufficiently large number of independent random variables is approximately normally distributed. The instructor also discusses the Student's t-distribution, particularly in the context of estimating the mean of a normally distributed population with small sample sizes. Throughout the lecture, visualizations are used to illustrate the concepts, allowing students to interactively explore the distributions. The session concludes with a discussion on the implications of these distributions in data science and statistical modeling, reinforcing the foundational knowledge necessary for further studies in statistics and data analysis.