ⓘ Convolution of probability distributions
The convolution of probability distributions arises in probability theory and statistics as the operation in terms of probability distributions that corresponds to the addition of independent random variables and, by extension, to forming linear combinations of random variables. The operation here is a special case of convolution in the context of probability distributions.
1. Introduction
The probability distribution of the sum of two or more independent random variables 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: see List of convolutions of probability distributions
The general formula for the distribution of the sum Z = X + Y {\displaystyle Z=X+Y} of two independent integervalued and hence discrete random variables is
P Z = z = ∑ k = − ∞ ∞ P X = k P Y = z − k {\displaystyle PZ=z=\sum _{k=\infty }^{\infty }PX=kPY=zk}The counterpart for independent continuously distributed random variables with density functions f, g {\displaystyle f,g} is
h z = f ∗ g z = ∫ − ∞ ∞ f z − t g t d t = ∫ − ∞ ∞ f t g z − t d t {\displaystyle hz=f*gz=\int _{\infty }^{\infty }fztgtdt=\int _{\infty }^{\infty }ftgztdt}If we start with random variables X and Y, related by Z=X+Y, and without knowledge of these random variables being independent, then:
f z = ∫ − ∞ ∞ f X Y x, z − x d x {\displaystyle f_{Z}z=\int \limits _{\infty }^{\infty }f_{XY}x,zx~dx}However, if X and Y are independent, then:
f X Y x, y = f x f y {\displaystyle f_{XY}x,y=f_{X}xf_{Y}y}and this formula becomes the convolution of probability distributions:
f z = ∫ − ∞ ∞ f x f Y z − x d x {\displaystyle f_{Z}z=\int \limits _{\infty }^{\infty }f_{X}x~f_{Y}zx~dx}2. Example derivation
There are several ways of deriving formulae for the convolution of probability distributions. Often the manipulation of integrals can be avoided by use of some type of generating function. Such methods can also be useful in deriving properties of the resulting distribution, such as moments, even if an explicit formula for the distribution itself cannot be derived.
One of the straightforward techniques is to use characteristic functions, which always exists and are unique to a given distribution.
Convolution of Bernoulli distributions
The convolution of two independent identically distributed Bernoulli random variables is a Binomial random variable. That is, in a shorthand notation,
∑ i = 1 2 B e r n o u l i p ∼ B i n o m i a l 2, p {\displaystyle \sum _{i=1}^{2}\mathrm {Bernoulli} p\sim \mathrm {Binomial} 2,p}To show this let
X i ∼ B e r n o u l i p, 0 < p < 1, 1 ≤ i ≤ 2 {\displaystyle X_{i}\sim \mathrm {Bernoulli} p,\quad 0 distribution of the sum is the convolution of the distributions of the individual random variables Consider the problem of generating a random variable
 same distribution up to location and scale parameters. The distributions of random variables having this property are said to be stable distributions
 density of a standard Cauchy distribution Density estimation Kernel density estimation Likelihood function List of probability distributions Probability mass
 factorization of distributions says that every probability distribution P admits in the convolution semi  group of probability distributions a factorization
 quasiprobability distributions also counterintuitively have regions of negative probability density, contradicting the first axiom. Quasiprobability distributions arise
 distributed. Many families of well  known infinitely divisible distributions are so  called convolution  closed, i.e. if the distribution of a Levy process at one
 In probability and statistics, a mixture distribution is the probability distribution of a random variable that is derived from a collection of other random
 the lattice of all partitions of that set. Random matrix Wigner semicircle distribution Circular law Free convolution Speicher, Roland 1994 Multiplicative
 Poisson, is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if
 Convolution is the operation of finding the probability distribution of a sum of independent random variables specified by probability distributions
 in probability theory List of probability distributions List of convolutions of probability distributions Glossary of experimental design Glossary of probability
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