List of convolutions of probability distributions explained

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

\sum ^{n} _{i =1} X _{i} \sim Y where X _{1,} X _{2,...,} X _{n} are independent random variables, and Y is the distribution that results from the convolution of X _{1,} X _{2,...,} X _{n} . In place of X _{i} and Y the names of the corresponding distributions and their parameters have been indicated.

Discrete distributions

\sum ^{n} _{i =1} Bernoulli (p) \sim Binomial (n,p) 0 <p< 1 n =1 , 2 , ... \sum ^{n} _{i =1} Binomial (n _{i,p)} \sim Binomial \left(\sum ^{n} _{i =1} n _{i,p\right)} 0 <p< 1 n _{i=1,2,...} \sum ^{n} _{i =1} NegativeBinomial (n _{i,p)} \sim NegativeBinomial \left(\sum ^{n} _{i =1} n _{i,p\right)} 0 <p< 1 n _{i=1,2,...} \sum ^{n} _{i =1} Geometric (p) \sim NegativeBinomial (n,p) 0 <p< 1 n =1 , 2 , ... \sum ^{n} _{i =1} Poisson ( λ _{i)} \sim Poisson \left(\sum ^{n} _{i =1} λ _{i\right)} λ _{i>0}

Continuous distributions

\sum ^{n} _{i =1} \operatorname{Stable}\left(\alpha,\beta _{i,c} _{i,\mu} ^{i\right) = \operatorname{Stable}\left(\alpha, (\sum ^{n} _{i =1} \beta _{i} c ^{\alpha} _{i})/(\sum ^{n} _{i =1} c ^{\alpha} _{i})} ,\left( \sum ^{n} _{i =1} c ^{\alpha} _{i} \right) ^{1/\alpha} ,\sum ^{n\mu} _{i\right)} 0 <\alpha _{i\le} 2 -1 \le \beta _{i} \le 1 c _{i>0} inf ty<\mu _{i<infty}

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

\sum ^{n} _{i =1} \operatorname{Normal}(\mu _{i,\sigma} ^{2 )} ^{i} \sim \operatorname{Normal}\left(\sum ^{n} _{i =1} \mu _{i,} \sum ^{n} _{i =1} \sigma ^{2 \right)} _{i} -inf ty<\mu _{i<infty} \sigma ^{2 > 0} _{i} (\alpha =2 , \beta _{i=0)} \sum ^{n} _{i =1} \operatorname{Cauchy}(a _{i,\gamma} _{i)} \sim \operatorname{Cauchy}\left(\sum ^{n} _{i =1} a _{i,} \sum ^{n} _{i =1} \gamma _{i\right)} -inf ty<a _{i<infty} \gamma _{i>0} (\alpha =1 , \beta _{i=0)} \sum ^{n} _{i =1} \operatorname{Levy}(\mu _{i,c} _{i)} \sim \operatorname{Levy}\left(\sum ^{n} _{i =1} \mu _{i,} \left(\sum ^{n} _{i =1} \sqrt{c ^{2 \right)} _{i}\right)} -inf ty<\mu _{i<infty} c _{i>0} (\alpha =1 / 2 , \beta _{i=1)} \sum ^{n} _{i =1} \operatorname{Gamma}(\alpha _{i,\beta)} \sim \operatorname{Gamma}\left(\sum ^{n} _{i =1} \alpha _{i,\beta\right)} \alpha _{i>0} \beta> 0 \sum ^{n} _{i =1} \operatorname{Voigt}(\mu _{i,\gamma} _{i,\sigma} _{i)} \sim \operatorname{Voigt}\left(\sum ^{n} _{i =1} \mu _{i,\sum} ^{n} ^{i =1} \gamma _{i,\sqrt{\sum} ^{n} ^{i =1} \sigma ^{2 }\right)} _{i} -inf ty<\mu _{i<infty} \gamma _{i>0} \sigma _{i>0} ^[1] \sum ^{n} _{i =1} \operatorname{VarianceGamma}(\mu _{i,\alpha,\beta,λ} _{i)} \sim \operatorname{VarianceGamma}\left(\sum ^{n} _{i =1} \mu _{i,} \alpha,\beta, \sum ^{n} _{i =1} λ _{i\right)} -inf ty<\mu _{i<infty} λ _{i} > 0 \sqrt{\alpha ^{2} - \beta ^{2}} > 0 ^[2] \sum ^{n} _{i =1} \operatorname{Exponential}(\theta) \sim \operatorname{Erlang}(n,\theta) \theta> 0 n =1 , 2 , ... \sum ^{n} _{i =1} \operatorname{Exponential}( λ _{i)} \sim \operatorname{Hypoexponential}( λ _{1,...,λ} _{n)} λ _{i>0} ^[3] \sum ^{n} _{i =1} \chi ^{2 (r} _{i)} \sim \chi ^{n} _{i =1} r _{i\right)} r _{i=1,2,...} \sum ^{r} _{i =1} N ^{2(0,1)} \sim \chi ^{2} _{r} r =1 , 2 , ... \sum ^{n(X} _{i} - \bar X) ^{2} \sim \sigma ^{2} \chi ^{2} _{n -1} , where X _{1,...,X} _{n} is a random sample from N(\mu,\sigma ^{2)} and \bar X = (1)/(n) \sum ^{n} _{i =1} X _{i.}

Mixed distributions:

\operatorname{Normal}(\mu,\sigma ^{2 ) + \operatorname{Cauchy}(x} _{0 ,\gamma)} \sim \operatorname{Voigt}(\mu + x _{0,\sigma,\gamma)} -inf ty<\mu< inf ty -inf ty<x _{0<infty} \gamma> 0 \sigma> 0

See also

• 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

Sources

• Book: Hogg . Robert V. . Robert V. Hogg . McKean . Joseph W. . Craig . Allen T. . Introduction to mathematical statistics . 6th . Prentice Hall . Upper Saddle River, New Jersey . 2004 . 692 . 978-0-13-008507-8 . 467974.

Notes and References

1. Web site: 2016 . 2012 . VoigtDistribution . 2021-04-08 . Wolfram Language Documentation.
2. Web site: 2012 . VarianceGammaDistribution . 2021-04-09 . Wolfram Language Documentation . 2016.
3. 2012.08498 . George P. . Yanev . Exponential and Hypoexponential Distributions: Some Characterizations . Mathematics . 2020-12-15. 8 . 12 . 2207 . 10.3390/math8122207 . free .