Lecture 8: Random Variables and Their Distributions | Statistics 110

The binomial distribution, written as Bin(n, p), is a fundamental concept in statistics with two parameters: n (number of trials) and p (probability of success). There are three ways to understand it: as the number of successes in n independent Bernoulli p trials (success/failure), as a sum of indicator random variables (IID Bernoulli p trials), or through its Probability Mass Function (PMF).

A random variable (RV) is a function that assigns a numerical value to each outcome in a sample space. An event like 'X = 7' refers to the set of outcomes where the random variable X takes the value 7. The Cumulative Distribution Function (CDF), F(x) = P(X ≤ x), describes the probability that a random variable X takes a value less than or equal to x, applicable to both discrete and continuous random variables. The Probability Mass Function (PMF), P(X = a_j), is specific to discrete random variables and lists the probability for each possible discrete value a_j, where probabilities must be non-negative and sum to 1.

The binomial PMF satisfies the conditions for a valid PMF, as its sum over all possible values equals 1, demonstrated by the binomial theorem. The lecture re-examines the sum of two independent binomial random variables (X ~ Bin(n,p), Y ~ Bin(m,p)), showing that X+Y ~ Bin(n+m,p), using both the 'story' interpretation (sum of trials) and the 'sum of indicator random variables' method. The lecture also derives this result algebraically through convolution, demonstrating Vandermonde's Identity.

The hypergeometric distribution is introduced through an example of counting aces in a 5-card hand. This scenario differs from binomial because trials are not independent (sampling without replacement). The PMF for the number of aces (or 'white marbles' in a general example) is derived using combinations. This distribution is analogous to the 'elk problem' discussed earlier in the course, highlighting the importance of understanding the underlying story rather than just memorizing formulas.

The hypergeometric distribution applies to sampling without replacement, making the trials dependent, unlike the binomial distribution which assumes independent trials (sampling with replacement). The sum of probabilities for the hypergeometric PMF also equals 1, again relying on Vandermonde's Identity. The lecture concludes with graphical representations of CDFs for continuous and discrete random variables, illustrating how continuous CDFs are smooth curves rising from 0 to 1, while discrete CDFs are step functions with jumps at the possible values of the random variable.