List of convolutions of probability distributions

In probability theory, 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 independent 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

∑ i = 1 n X i ∼ Y {\displaystyle \sum _{i=1}^{n}X_{i}\sim Y}

where  X 1 , X 2 , … , X n {\displaystyle X_{1},X_{2},\dots ,X_{n}} are independent random variables, and  Y {\displaystyle Y} is the distribution that results from the convolution of  X 1 , X 2 , … , X n {\displaystyle X_{1},X_{2},\dots ,X_{n}} . In place of  X i {\displaystyle X_{i}} and  Y {\displaystyle Y} the names of the corresponding distributions and their parameters have been indicated.

## Discrete distributions

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* ∑ i = 1 n B e r n o u l l i ( p ) ∼ B i n o m i a l ( n , p ) 0 < p < 1 n = 1 , 2 , … {\displaystyle \sum _{i=1}^{n}\mathrm {Bernoulli} (p)\sim \mathrm {Binomial} (n,p)\qquad 0<p<1\quad n=1,2,\dots }
* ∑ i = 1 n B i n o m i a l ( n i , p ) ∼ B i n o m i a l ( ∑ i = 1 n n i , p ) 0 < p < 1 n i = 1 , 2 , … {\displaystyle \sum _{i=1}^{n}\mathrm {Binomial} (n_{i},p)\sim \mathrm {Binomial} \left(\sum _{i=1}^{n}n_{i},p\right)\qquad 0<p<1\quad n_{i}=1,2,\dots }
* ∑ i = 1 n N e g a t i v e B i n o m i a l ( n i , p ) ∼ N e g a t i v e B i n o m i a l ( ∑ i = 1 n n i , p ) 0 < p < 1 n i = 1 , 2 , … {\displaystyle \sum _{i=1}^{n}\mathrm {NegativeBinomial} (n_{i},p)\sim \mathrm {NegativeBinomial} \left(\sum _{i=1}^{n}n_{i},p\right)\qquad 0<p<1\quad n_{i}=1,2,\dots }
* ∑ i = 1 n G e o m e t r i c ( p ) ∼ N e g a t i v e B i n o m i a l ( n , p ) 0 < p < 1 n = 1 , 2 , … {\displaystyle \sum _{i=1}^{n}\mathrm {Geometric} (p)\sim \mathrm {NegativeBinomial} (n,p)\qquad 0<p<1\quad n=1,2,\dots }
* ∑ i = 1 n P o i s s o n ( λ i ) ∼ P o i s s o n ( ∑ i = 1 n λ i ) λ i > 0 {\displaystyle \sum _{i=1}^{n}\mathrm {Poisson} (\lambda _{i})\sim \mathrm {Poisson} \left(\sum _{i=1}^{n}\lambda _{i}\right)\qquad \lambda _{i}>0}

## Continuous distributions

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* ∑ i = 1 n Stable ⁡ ( α , β i , c i , μ i ) = Stable ⁡ ( α , ∑ i = 1 n β i c i α ∑ i = 1 n c i α , ( ∑ i = 1 n c i α ) 1 / α , ∑ i = 1 n μ i ) {\displaystyle \sum _{i=1}^{n}\operatorname {Stable} \left(\alpha ,\beta _{i},c_{i},\mu _{i}\right)=\operatorname {Stable} \left(\alpha ,{\frac {\sum _{i=1}^{n}\beta _{i}c_{i}^{\alpha }}{\sum _{i=1}^{n}c_{i}^{\alpha }}},\left(\sum _{i=1}^{n}c_{i}^{\alpha }\right)^{1/\alpha },\sum _{i=1}^{n}\mu _{i}\right)}

0 < α i ≤ 2 − 1 ≤ β i ≤ 1 c i > 0 ∞ < μ i < ∞ {\displaystyle \qquad 0<\alpha _{i}\leq 2\quad -1\leq \beta _{i}\leq 1\quad c_{i}>0\quad \infty <\mu _{i}<\infty }

The following three statements are special cases of the above statement:

* ∑ i = 1 n Normal ⁡ ( μ i , σ i 2 ) ∼ Normal ⁡ ( ∑ i = 1 n μ i , ∑ i = 1 n σ i 2 ) − ∞ < μ i < ∞ σ i 2 > 0 ( α = 2 , β i = 0 ) {\displaystyle \sum _{i=1}^{n}\operatorname {Normal} (\mu _{i},\sigma _{i}^{2})\sim \operatorname {Normal} \left(\sum _{i=1}^{n}\mu _{i},\sum _{i=1}^{n}\sigma _{i}^{2}\right)\qquad -\infty <\mu _{i}<\infty \quad \sigma _{i}^{2}>0\quad (\alpha =2,\beta _{i}=0)}
* ∑ i = 1 n Cauchy ⁡ ( a i , γ i ) ∼ Cauchy ⁡ ( ∑ i = 1 n a i , ∑ i = 1 n γ i ) − ∞ < a i < ∞ γ i > 0 ( α = 1 , β i = 0 ) {\displaystyle \sum _{i=1}^{n}\operatorname {Cauchy} (a_{i},\gamma _{i})\sim \operatorname {Cauchy} \left(\sum _{i=1}^{n}a_{i},\sum _{i=1}^{n}\gamma _{i}\right)\qquad -\infty <a_{i}<\infty \quad \gamma _{i}>0\quad (\alpha =1,\beta _{i}=0)}
* ∑ i = 1 n Levy ⁡ ( μ i , c i ) ∼ Levy ⁡ ( ∑ i = 1 n μ i , ( ∑ i = 1 n c i ) 2 ) − ∞ < μ i < ∞ c i > 0 ( α = 1 / 2 , β i = 1 ) {\displaystyle \sum _{i=1}^{n}\operatorname {Levy} (\mu _{i},c_{i})\sim \operatorname {Levy} \left(\sum _{i=1}^{n}\mu _{i},\left(\sum _{i=1}^{n}{\sqrt {c_{i}}}\right)^{2}\right)\qquad -\infty <\mu _{i}<\infty \quad c_{i}>0\quad (\alpha =1/2,\beta _{i}=1)}

* ∑ i = 1 n Gamma ⁡ ( α i , β ) ∼ Gamma ⁡ ( ∑ i = 1 n α i , β ) α i > 0 β > 0 {\displaystyle \sum _{i=1}^{n}\operatorname {Gamma} (\alpha _{i},\beta )\sim \operatorname {Gamma} \left(\sum _{i=1}^{n}\alpha _{i},\beta \right)\qquad \alpha _{i}>0\quad \beta >0}
* ∑ i = 1 n Voigt ⁡ ( μ i , γ i , σ i ) ∼ Voigt ⁡ ( ∑ i = 1 n μ i , ∑ i = 1 n γ i , ∑ i = 1 n σ i 2 ) − ∞ < μ i < ∞ γ i > 0 σ i > 0 {\displaystyle \sum _{i=1}^{n}\operatorname {Voigt} (\mu _{i},\gamma _{i},\sigma _{i})\sim \operatorname {Voigt} \left(\sum _{i=1}^{n}\mu _{i},\sum _{i=1}^{n}\gamma _{i},{\sqrt {\sum _{i=1}^{n}\sigma _{i}^{2}}}\right)\qquad -\infty <\mu _{i}<\infty \quad \gamma _{i}>0\quad \sigma _{i}>0} [1]
* ∑ i = 1 n VarianceGamma ⁡ ( μ i , α , β , λ i ) ∼ VarianceGamma ⁡ ( ∑ i = 1 n μ i , α , β , ∑ i = 1 n λ i ) − ∞ < μ i < ∞ λ i > 0 α 2 − β 2 > 0 {\displaystyle \sum _{i=1}^{n}\operatorname {VarianceGamma} (\mu _{i},\alpha ,\beta ,\lambda _{i})\sim \operatorname {VarianceGamma} \left(\sum _{i=1}^{n}\mu _{i},\alpha ,\beta ,\sum _{i=1}^{n}\lambda _{i}\right)\qquad -\infty <\mu _{i}<\infty \quad \lambda _{i}>0\quad {\sqrt {\alpha ^{2}-\beta ^{2}}}>0} [2]
* ∑ i = 1 n Exponential ⁡ ( θ ) ∼ Erlang ⁡ ( n , θ ) θ > 0 n = 1 , 2 , … {\displaystyle \sum _{i=1}^{n}\operatorname {Exponential} (\theta )\sim \operatorname {Erlang} (n,\theta )\qquad \theta >0\quad n=1,2,\dots }
* ∑ i = 1 n Exponential ⁡ ( λ i ) ∼ Hypoexponential ⁡ ( λ 1 , … , λ n ) λ i > 0 {\displaystyle \sum _{i=1}^{n}\operatorname {Exponential} (\lambda _{i})\sim \operatorname {Hypoexponential} (\lambda _{1},\dots ,\lambda _{n})\qquad \lambda _{i}>0} [3]
* ∑ i = 1 n χ 2 ( r i ) ∼ χ 2 ( ∑ i = 1 n r i ) r i = 1 , 2 , … {\displaystyle \sum _{i=1}^{n}\chi ^{2}(r_{i})\sim \chi ^{2}\left(\sum _{i=1}^{n}r_{i}\right)\qquad r_{i}=1,2,\dots }
* ∑ i = 1 r N 2 ( 0 , 1 ) ∼ χ r 2 r = 1 , 2 , … {\displaystyle \sum _{i=1}^{r}N^{2}(0,1)\sim \chi _{r}^{2}\qquad r=1,2,\dots }
* ∑ i = 1 n ( X i − X ¯ ) 2 ∼ σ 2 χ n − 1 2 , {\displaystyle \sum _{i=1}^{n}(X_{i}-{\bar {X}})^{2}\sim \sigma ^{2}\chi _{n-1}^{2},\quad } where  X 1 , … , X n {\displaystyle X_{1},\dots ,X_{n}} is a random sample from  N ( μ , σ 2 ) {\displaystyle N(\mu ,\sigma ^{2})} and  X ¯ = 1 n ∑ i = 1 n X i . {\displaystyle {\bar {X}}={\frac {1}{n}}\sum _{i=1}^{n}X_{i}.}

Mixed distributions:

* Normal ⁡ ( μ , σ 2 ) + Cauchy ⁡ ( x 0 , γ ) ∼ Voigt ⁡ ( μ + x 0 , σ , γ ) − ∞ < μ < ∞ − ∞ < x 0 < ∞ γ > 0 σ > 0 {\displaystyle \operatorname {Normal} (\mu ,\sigma ^{2})+\operatorname {Cauchy} (x_{0},\gamma )\sim \operatorname {Voigt} (\mu +x_{0},\sigma ,\gamma )\qquad -\infty <\mu <\infty \quad -\infty <x_{0}<\infty \quad \gamma >0\quad \sigma >0}

## See also

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* Algebra of random variables
* Relationships among probability distributions
* Infinite divisibility (probability)
* Bernoulli distribution
* Binomial distribution
* Cauchy distribution
* Erlang distribution
* Exponential distribution
* Gamma distribution
* Geometric distribution
* Hypoexponential distribution
* Lévy distribution
* Poisson distribution
* Stable distribution
* Mixture distribution
* Sum of normally distributed random variables

## References

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1. ^ "VoigtDistribution". Wolfram Language Documentation. 2016 [2012]. Retrieved 2021-04-08.
2. ^ "VarianceGammaDistribution". Wolfram Language Documentation (published 2016). 2012. Retrieved 2021-04-09.
3. ^ Yanev, George P. (2020-12-15). "Exponential and Hypoexponential Distributions: Some Characterizations". Mathematics. 8 (12): 2207. arXiv:2012.08498. doi:10.3390/math8122207.

## Sources

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* Hogg, Robert V.; McKean, Joseph W.; Craig, Allen T. (2004). Introduction to mathematical statistics (6th ed.). Upper Saddle River, New Jersey: Prentice Hall. p. 692\. ISBN 978-0-13-008507-8. MR 0467974.