Probability Distributions: Central Limit Theorem and Applications

This lecture covers essential concepts in statistics, focusing on probability distributions and the Central Limit Theorem (CLT). The instructor explains that the CLT states that the sample mean of a sufficiently large number of independent and identically distributed (i.i.d.) random variables is approximately normally distributed. The larger the sample size, the better the approximation. The lecture distinguishes between discrete and continuous probability distributions, highlighting key examples such as the normal distribution, exponential distribution, and Student's t-distribution. The instructor emphasizes the importance of understanding these distributions in the context of data science and statistical analysis. Visualizations are used to illustrate the probability density functions and cumulative distribution functions of various distributions, allowing students to interactively explore how changes in parameters affect the distributions. The lecture concludes with a discussion on the practical applications of these statistical concepts in real-world scenarios, reinforcing the relevance of probability distributions in data science and decision-making processes.