# 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

Obtained from the sum of independent Bernoulli random variables

Takes value 1 when an experiment succeeds and 0 otherwise

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

The distribution of the number of trials needed to get a success from repeated Bernoulli experiments

## Univariate continuous probability distributions

Most commonly used to model waiting times

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

Obtained from the sum of squared normal random variables

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

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

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

Used to model uncertainty about proportions and probabilities of binomial outcomes

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

The distribution of the exponential of a normal random variable

## Multivariate discrete probability distributions

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

A multivariate generalization of the Bernoulli distribution

## Multivariate continuous probability distributions

A multivariate generalization of Student's t distribution

A multivariate generalization of the normal distribution

Generalizes the Gamma distribution to random matrices

## Transformations of multivariate normal vectors

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

Linear transformations of normal vectors preserve normality

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

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