List of convolutions of probability distributions
Encyclopedia
In probability theory
, the probability distribution
of the sum of two or more independent random variable
s is the convolution of their individual distributions. The term is motivated by the fact that the probability mass function
or probability density function
of a sum of random variables is the convolution
of their corresponding probability mass functions or probability density functions respectively. Many well known distributions have simple convolutions. The following is a list of these convolutions. Each statement is of the form
where
are independent and identically distributed random variables. In place of 
and 
the names of the corresponding distributions and their parameters have been indicated.
Probability theory
Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...
, the probability distribution
Probability distribution
In probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....
of the sum of two or more independent random variable
Random variable
In probability and statistics, a random variable or stochastic variable is, roughly speaking, a variable whose value results from a measurement on some type of random process. Formally, it is a function from a probability space, typically to the real numbers, which is measurable functionmeasurable...
s is the convolution of their individual distributions. The term is motivated by the fact that the probability mass function
Probability mass function
In probability theory and statistics, a probability mass function is a function that gives the probability that a discrete random variable is exactly equal to some value...
or probability density function
Probability density function
In probability theory, a probability density function , or density of a continuous random variable is a function that describes the relative likelihood for this random variable to occur at a given point. The probability for the random variable to fall within a particular region is given by the...
of a sum of random variables is the convolution
Convolution
In mathematics and, in particular, functional analysis, convolution is a mathematical operation on two functions f and g, producing a third function that is typically viewed as a modified version of one of the original functions. Convolution is similar to cross-correlation...
of their corresponding probability mass functions or probability density functions respectively. Many well known distributions have simple convolutions. The following is a list of these convolutions. Each statement is of the form
where
are independent and identically distributed random variables. In place of and the names of the corresponding distributions and their parameters have been indicated.
