Probability

This course introduces students who have taken multivariable calculus to probability, the mathematical foundation of statistics. It begins with the intuitive idea of\ud the probability of an outcome in a (repeatable) experiment. It teaches students elementary combinatorics so that they can compute probabilities when the experiment has only finitely many possible outcomes, and introduces random variables in this context. (One sixth is the probability that the sum X of two die rolls equals 7.) It then uses calculus to build on this understanding of discrete random variables X to explain probability density functions, cumulative distribution functions, and joint distribution functions. (A scatterplot in the plane is a sample from the joint distribution of a pair of continuous random variables, X and Y .) Thereby students gain a deep understanding of the role that probability theory plays in multivariate statistical analysis. To clarify the numerous applications of probability distributions, the course covers topics such as the expected value and variance of a random variable; the covariance of a pair of random variables; and a limit theorem about sums of more and more independently-chosen values from a fixed distribution (the Central Limit Theorem). Finally, we study how to use data to estimate parameters of a distribution (like the mean and variance of a normal distribution), and how to compare estimator