Use these tables to revise the various connections between probability distributions.

In the following relations the starting distribution is a univariate discrete probability distribution.

Start from distribution | Do | Obtain | Find proof |
---|---|---|---|

Bernoulli distribution | Sum of independent Bernoulli random variables. | Binomial distribution | * |

Observe a sequence of realizations of independent Bernoulli random variables and record the number of 0s obtained before a 1 shows up. | Geometric distribution | * | |

Assign a Beta prior to the probability that a Bernoulli variable will be equal to 1. Then, observe a realization of the Bernoulli variable and update the prior. | Beta distribution | * | |

Binomial distribution | Set parameter for number of trials equal to 1. | Bernoulli distribution | * |

The most common univariate continuous distributions have lots of interesting relationships with other distributions.

Start from distribution | Do | Obtain | Find proof |
---|---|---|---|

Exponential distribution | Sum of independent exponential random variables with common rate parameter. | Gamma distribution | * |

Keep summing the realizations of independent exponential random variables while the sum is less than 1. Record the number of variables you have summed. | Poisson distribution | * | |

Normal distribution | Take n mutually independent standard normal variables, square them and sum the squares. | Chi-square distribution | * |

Take n mutually independent zero-mean normal variables, square them and sum the squares. | Gamma distribution | * | |

Take a linear combination of n mutually independent normal variables. | Normal distribution | * | |

Add a constant and/or multiply by a constant. | Normal distribution | * | |

Take a standard normal variable Z, independent of a variable X having a chi-square distribution with n degrees of freedom. Then, compute Z/sqrt(X/n). | Student's t distribution | * | |

Take the exponential. | Log-normal distribution | * | |

Collect n independent normal random variables in a vector. | Multivariate normal distribution | * | |

Chi-square distribution | Sum two independent chi-square random variables. | Chi-square distribution | * |

Multiply by a strictly positive constant. | Gamma distribution | * | |

Take a chi-square variable X with n degrees of freedom, independent of a standard normal variable Z. Then, compute Z/sqrt(X/n). | Student's t distribution | * | |

Take a chi-square variable X1 with n1 degrees of freedom, independent of another chi-square variable X2 with n2 degrees of freedom. Then, compute (X1/n1)/(X2/n2). | F distribution | * | |

Before drawing a vector from a multivariate normal distribution, divide the covariance matrix by an independent chi-square variable. | Multivariate t distribution | * | |

Gamma distribution | Multiply by a strictly positive constant. | Gamma distribution | * |

Set the two parameters in a certain special manner (depending on parametrization). | Chi-quare distribution | * | |

Take a Gamma random variable G, independent of a zero-mean normal variable Z. Then, compute Z/sqrt(G). | Student's t distribution | * | |

Before drawing a vector from a multivariate normal distribution, divide the covariance matrix by an independent Gamma random variable. | Multivariate t distribution | * | |

Take several independent Gammas and divide them by their sum. | Dirichlet distribution | * | |

Student's t distribution | Let the degrees-of-freedom parameter go to infinity. | Normal distribution | * |

Beta distribution | Set the two parameters equal to 1. | Uniform distribution | * |

Assign a Beta prior to the probability of success in a Bernoulli experiment. Then, observe the outcome of the experiment and update the prior. | Beta distribution | * | |

Log-normal distribution | Take the natural logarithm. | Normal distribution | * |

Find out how multivariate discrete probability distributions are linked to other distributions.

Start from distribution | Do | Obtain | Find proof |
---|---|---|---|

Multinoulli distribution | Compute the marginal distribution of one entry of a Multinoulli random vector. | Bernoulli distribution | * |

Sum n iid Multinoulli vectors. | Multinomial distribution | * | |

Multinomial distribution | Set the number-of-trials parameter equal to 1. | Multinoulli distribution | * |

Compute the marginal distribution of one entry of a multinomial random vector. | Binomial distribution | * |

Finally, discover the relations between multivariate continuous distributions and other probability distributions.

Start from distribution | Do | Obtain | Find proof |
---|---|---|---|

Multivariate normal distribution | Set the number of entries of a multivariate normal random vector equal to 1. | Normal distribution | * |

Compute the marginal distribution of one of the entries of a multivariate normal random vector. | Normal distribution | ||

Take a linear transformation. | Multivariate normal distribution | * | |

Sum n independent multivariate normal random vectors. | Multivariate normal distribution | * | |

Build a quadratic form using a multivariate normal random vector and a symmetric idempotent matrix. | Chi-square distribution | * | |

Sum of outer products of n iid zero-mean multivariate normal random vectors. | Wishart distribution | * | |

Before drawing a vector from a multivariate normal distribution, divide the covariance matrix by an independent chi-square variable. | Multivariate t distribution | * | |

Before drawing a vector from a multivariate normal distribution, divide the covariance matrix by an independent Gamma random variable. | Multivariate t distribution | * | |

Multivariate t distribution | Set the number of entries of a multivariate t vector equal to 1. | Student's t distribution | * |

Compute the marginal distribution of one entry of a multivariate t vector. | Student's t distribution | * | |

Dirichlet distribution | Set the number of entries of a Dirichlet vector equal to 1. | Beta distribution | * |

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