StatlectThe Digital Textbook

Probability distributions

This compendium contains a list of some of the most frequently encountered probability distributions. Follow the links to access detailed derivations of the main properties and solved exercises.

Univariate discrete probability distributions

Binomial distribution

Obtained from the sum of independent Bernoulli random variables

Bernoulli distribution

Takes value 1 when an experiment succeeds and 0 otherwise

Poisson distribution

Used to model the number of unpredictable events within a unit of time

Univariate continuous probability distributions

Exponential distribution

Most commonly used to model waiting times

Uniform distribution

Assigns the same probability to intervals having the same length and belonging to its support

Chi-square distribution

Obtained from the sum of squared normal random variables

Normal distribution

The infamous bell-shaped distribution, used to model a variety of natural and social phenomena

Student's t-distribution

Obtained from the ratio of a normal random variable to the square root of a Gamma

Gamma distribution

Obtained from the product of a Chi-square random variable and a positive constant

Beta distribution

Used to model uncertainty about proportions and probabilities of binomial outcomes

F distribution

The ratio between two Chi-square random variables, divided by their degrees of freedom

Multivariate discrete probability distributions

Multinomial distribution

Generalizes the binomial distribution to the case of more than two outcomes

Multinoulli distribution

A multivariate generalization of the Bernoulli distribution

Multivariate continuous probability distributions

Multivariate Student's t distribution

A multivariate generalization of Student's t distribution

Multivariate normal distribution

A multivariate generalization of the normal distribution

Wishart distribution

Generalizes the Gamma distribution to random matrices

Transformations of multivariate normal vectors

Quadratic forms involving normal vectors

Certain quadratic forms involving normal vectors have a Chi-square distribution

Linear transformations of normal vectors

Linear transformations of normal vectors preserve normality

Partitions of normal vectors

Normality and independence of the sub-vectors of a normal vector

The book

Most learning materials found on this website are now available in a traditional textbook format.