Mathematics · Ch 11 — Probability Distributions
Introduction
Introduction
In Volume 2 of the Class 11 course, a sample space was built to completely describe the possible outcomes of a random experiment — for a coin toss, ; for a die, ; and so on. These outcomes are not always numbers (a head or a tail is not a number), yet almost everything we want to compute about an experiment — averages, spreads, decision rules — is easiest to express through numbers.
This chapter studies exactly that bridge: a function, called a random variable, that is defined on the sample space of a random experiment and assigns a real number to every outcome. Once outcomes have been turned into numbers, we study their probability distribution — how the total probability of is spread across the possible numerical values.
The chapter's roadmap: define a random variable and its two flavours — discrete (countable values, e.g. number of heads) and continuous (an interval of values, e.g. a lifetime) — then, for each flavour, learn the probability mass/density function, the cumulative distribution function, how to pass between the two, and finally mean, variance, and the two most important named discrete distributions: the Bernoulli and binomial distributions.