Relationships among probability distributions

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

Univariate discrete 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	*

Univariate continuous distributions

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	*

Multivariate discrete distributions

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	*

Multivariate continuous distributions

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	*