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.
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
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
Generalizes the binomial distribution to the case of more than two outcomes
A multivariate generalization of the Bernoulli distribution
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
Multivariate generalization of the Beta distribution used for vectors of random probabilities
Generalizes the Gamma distribution to random matrices
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
Normality and independence of the sub-vectors of a normal vector
Chi-square distribution values
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
Relationships among distributions
Review the various connections among the probability distributions in this list
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 |
Most of the learning materials found on this website are now available in a traditional textbook format.
