3: Distributions

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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 sections, we study convergence in distribution, one of the most important types of convergence. We also construct the abstract integral with respect to a positive measure and study the basic properties of the integral. This leads in turn to general (signed measures), absolute continuity and singularity, and the existence of density functions. Finally, we study various vector spaces of functions that are defined by integral pro

3.1: Discrete Distributions
3.2: Continuous Distributions
In the previous section, we considered discrete distributions. In this section, we study a complementary type of distribution. As usual, if you are a new student of probability, you may want to skip the technical details.
3.3: Mixed Distributions
In the previous two sections, we studied discrete probability meausres and continuous probability measures. In this section, we will study probability measure that that are combinations of the two types. Once again, if you are a new student of probability you may want to skip the technical details.
3.4: Joint Distributions
The purpose of this section is to study how the distribution of a pair of random variables is related to the distributions of the variables individually.
3.5: Conditional Distributions
In this section, we study how a probability distribution changes when a given random variable has a known, specified value. So this is an essential topic that deals with hou probability measures should be updated in light of new information. As usual, if you are a new student or probability, you may want to skip the technical details.
3.6: Distribution and Quantile Functions
3.7: Transformations of Random Variables
3.8: Convergence in Distribution
3.9: General Distribution Functions
3.10: The Integral With Respect to a Measure
3.11: Properties of the Integral
3.12: General Measures
3.13: Absolute Continuity and Density Functions
3.14: Function Spaces