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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

Binomial distribution

Obtained as 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

Geometric distribution

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

The number of repeated Bernoulli trials performed before obtaining a success has a geometric distribution.

Univariate continuous probability distributions

Exponential distribution

This probability distribution is 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

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

Normal distribution

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

Student's t-distribution

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

Gamma distribution

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

Log-normal distribution

The distribution of the exponential of a normal random variable

If you multiply a Chi-square random variable by a positive constant, you get...

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

Different ways to encode a binomial variable: the Bernoulli and multinoulli distributions.

Multivariate continuous probability distributions

Multivariate Student's t distribution

A multivariate generalization of the Student's t distribution

Multivariate normal distribution

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

Dirichlet distribution

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

Wishart distribution

Generalizes the Gamma distribution to random matrices

If you divide a multivariate normal random vector by the square root of a Gamma random variable, you get a multivariate Student distribution.

Transformations of multivariate normal vectors

Quadratic forms involving normal vectors

Quadratic forms involving normal vectors, often found in statistics, 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

A quadratic form in a standard normal random vector has a chi-square distribution.

Other topics

Chi-square distribution values

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

Normal distribution values

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

Relationships among distributions

Review the various connections among the probability distributions in this list

We also have a list of common probability distributions for random matrices.

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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