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 |

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- Convergence in probability
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