# Probability distributions

This is a list of probability distributions commonly used in statistics. For each distribution you will find explanations, examples and a problem set with solved exercises.

## Univariate discrete probability distributions

Obtained as 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

This probability distribution is most commonly used to model waiting times

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

The sum of squared normal random variables often pops up in statistics

The most famous distribution in the list, used to model a variety of natural and social phenomena

The ratio of a normal random variable to the square root of a Gamma

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 the Student's t distribution

A multivariate generalization of the normal distribution, frequently used in statistics

Multivariate generalization of the Beta distribution used for vectors of random probabilities

Generalizes the Gamma distribution to random matrices

## Transformations of multivariate normal vectors

Quadratic forms involving normal vectors, often found in statistics, have a Chi-square distribution

Linear transformations of normal vectors preserve normality

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

## Other topics

Examples of how to find the values of the cumulative distribution function of a chi-square variable

This lecture explains how to find the values of the cumulative distribution function of a normal variable

Review the various connections among the probability distributions in this list

## How to define the term "probability distribution"

Did you know that the term "probability distribution" is often used loosely, without a precise mathematical meaning?

The term may refer to any one of the functions used to assign probabilities to the sets of values that a random variable can take.

Here is a list of the most common functions.

Name of function Variable/vector Type of distribution
Cumulative distribution function Variable All
Probability mass function Variable Discrete
Probability density function Variable Continuous
Characteristic function Variable All
Moment generating function Variable Only some of those with finite moments
Joint distribution function Vector All
Joint probability mass function Vector Discrete
Joint probability density function Vector Continuous
Joint characteristic function Vector All
Joint moment generating function Vector Only some of those with finite cross-moments
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