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Convolution of probability distributions

Theconvolution of probability distributions arises in

probability theoryandstatisticsas the operation in terms

ofprobability distributionsthat corresponds to the addi-

tion ofindependent random variablesand, by extension,

to forming linear combinations of random variables. The

operation here is a special case ofconvolutionin the con-

text of probability distributions.

1 Introduction

Theprobability distributionof the sum of two or more

independent random variablesis 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 func-

tions or probability density functions respectively. Many

well known distributions have simple convolutions: see

List of convolutions of probability distributions

2 Example derivation

There are several ways of deriving formulae for the con-

volution of probability distributions. Often the manipu-

lation 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.

2.1 Convolution of Bernoulli distributions

The convolution of two i.i.d.Bernoulli random variables

is a Binomial random variable. That is, in a shorthand

notation,

2i=1

Bernoulli(p) Binomial(2, p).

To show this let

Xi Bernoulli(p), 0< p n in the

last but three equality, and ofPascals rulein the second

last equality.

2.1.2 Using characteristic functions

The characteristic function of eachXk and ofZis

Xk(t) = 1 p +peit

Z(t) =

1 p +peit2

where tis within someneighborhoodof zero.

1

https://en.wikipedia.org/wiki/Neighborhood_(mathematics)https://en.wikipedia.org/wiki/Pascal%2527s_rulehttps://en.wikipedia.org/wiki/Bernoulli_distributionhttps://en.wikipedia.org/wiki/Characteristic_function_(probability_theory)https://en.wikipedia.org/wiki/Generating_functionhttps://en.wikipedia.org/wiki/List_of_convolutions_of_probability_distributionshttps://en.wikipedia.org/wiki/Convolutionhttps://en.wikipedia.org/wiki/Probability_density_functionhttps://en.wikipedia.org/wiki/Probability_density_functionhttps://en.wikipedia.org/wiki/Probability_mass_functionhttps://en.wikipedia.org/wiki/Random_variablehttps://en.wikipedia.org/wiki/Independent_(probability)https://en.wikipedia.org/wiki/Probability_distributionhttps://en.wikipedia.org/wiki/Convolutionhttps://en.wikipedia.org/wiki/Random_variablehttps://en.wikipedia.org/wiki/Statistically_independenthttps://en.wikipedia.org/wiki/Probability_distributionhttps://en.wikipedia.org/wiki/Statisticshttps://en.wikipedia.org/wiki/Probability_theory

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2 4 REFERENCES

Y(t) =E

eit

2

k=1Xk

= E

2k=1

eitXk

=2

k=1

E eitXk=2

k=1

1 p +peit=

1 p +peit2

=Z(t)

Theexpectationof the product is the product of the ex-

pectations since eachXk is independent. SinceY andZ

have the same characteristic function, they must have the

same distribution.

3 See also

List of convolutions of probability distributions

4 References

Hogg, Robert V.; McKean, Joseph W.; Craig, Allen

T. (2004). Introduction to mathematical statistics

(6th ed.). Upper Saddle River, New Jersey: Pren-

tice Hall. p. 692. ISBN 978-0-13-008507-8. MR

467974.

https://www.ams.org/mathscinet-getitem?mr=467974https://en.wikipedia.org/wiki/Mathematical_Reviewshttps://en.wikipedia.org/wiki/Special:BookSources/978-0-13-008507-8https://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://www.pearsonhighered.com/educator/product/Introduction-to-Mathematical-Statistics/9780130085078.pagehttps://en.wikipedia.org/wiki/Robert_V._Hogghttps://en.wikipedia.org/wiki/List_of_convolutions_of_probability_distributionshttps://en.wikipedia.org/wiki/Expected_value

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