Back

ⓘ Convolution of probability distributions. The convolution of probability distributions arises in probability theory and statistics as the operation in terms of ..




                                     

ⓘ 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 integer-valued 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=z-k}

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 }fz-tgtdt=\int _{-\infty }^{\infty }ftgz-tdt}

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,z-x~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}z-x~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

Users also searched:

convolution of distributions, sum of random variables,

...
...
...