StatlectThe Digital Textbook
Index

Fundamentals of probability

This is an introduction to the main concepts of probability theory. Each lecture contains detailed proofs and derivations of all the main results, as well as solved exercises.

Probability and events

Zero-probability events

Events having zero probability, almost sure events, almost sure properties

Probability

Sample space, sample points, events, probability and its properties

Bayes' rule

Prior probability, posterior probability, updating

Conditional probability

How to revise probabilities when new information arrives

Independent events

Definition and explanation of independence and mutual independence

Random variables and random vectors

Random vectors

Joint distributions, marginal distributions

Random variables

Discrete and continuous random variables, probability mass and density functions

Expected value and the Lebesgue integral

A rigorous definition of expected value, based on the Lebesgue integral

Expected value

The mean of a random variable, how to compute it, its properties

Variance

Dispersion around the mean, definition, interpretation, properties

Properties of the expected value

Linearity of the expected value, expectation of positive random variables, other properties

Linear correlation

Another measure of association between random variables

Covariance

Association between random variables, definition, interpretation, properties

Indicator functions

Equal to one when an event happens and zero otherwise

Covariance matrix

A multivariate generalization of the concept of variance

Quantile

Cut-off point of a distribution that leaves to its left a given proportion of the distribution

Conditional probabilities, conditional distributions and independence

Conditional probability distributions

How to update the distribution of a random variable after receiving some information

Conditional probability as a random variable

A more rigorous presentation of conditional probability

Independent random variables

Definition and characterization of independence between random variables

Conditional expectation

How to compute the expected value of a random variable after observing the value of another

Inequalities

Chebyshev's inequality

An important inequality derived from Markov's inequality

Markov's inequality

Provides an upper bound to the probability that a random variable will exceed a threshold

Jensen's inequality

Concerns the expected value of convex and concave transformations of a random variable

More about probability mass and density functions

Legitimate probability density functions

Properties of probability density functions and how to construct them

Legitimate probability mass functions

Properties of probability mass functions and how to construct them

Factorization of joint probability density functions

Factorization into marginal and conditional probability density function

Factorization of joint probability mass functions

Factorization into marginal and conditional probability mass function

Transformations of random variables

Functions of random vectors and their distribution

How to derive the joint distribution of a function of a random vector

Functions of random variables and their distribution

How to derive the distribution of Y=g(X) from the distribution of X

Sums of independent random variables

How to derive the distribution of a sum from the distributions of the summands

Moments

Cross-moments

Definition of cross-moment and central cross-moment of a random vector

Moments

Definition of moment and central moment of a random variable

Moment generating and characteristic functions

Joint moment generating function

Generalizes the concept of moment generating function to random vectors

Moment generating function

Definition, computation of moments, characterization of distributions

Joint characteristic function

Generalizes the concept of characteristic function to random vectors

Characteristic function

Definition, computation of moments, characterization of distributions

The book

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