Recall that a probability distribution is just another name for a probability measure. Most distributions are associated with random variables, and in fact every distribution can be associated with a random variable. In this chapter we explore the basic types of probability distributions (discrete, continuous, mixed), and the ways that distributions can be defined using density functions, distribution functions, and quantile functions. We also study the relationship between the distribution of a random vector and the distributions of its components, conditional distributions, and how the distribution of a random variable changes when the variable is transformed. In the advanced section, we study convergence in distribution, one of the most important types of convergence.
Nothing can permanently please which does not contain in itself the reason why it is so and not otherwise.—Samuel Taylor Coleridge