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

: $\sum_{i=1}^n X_i \sim Y$

where $X_1, X_2,\dots, X_n$ are independent random variables, and $Y$ is the distribution that results from the convolution of $X_1, X_2,\dots, X_n$ . In place of $X_i$ and $Y$ the names of the corresponding distributions and their parameters have been indicated.

Discrete distributions

$\sum_{i=1}^n \mathrm{Bernoulli}(p) \sim \mathrm{Binomial}(n,p) \qquad 0$
$\sum_{i=1}^n \mathrm{Binomial}(n_i,p) \sim \mathrm{Binomial}\left(\sum_{i=1}^n n_i,p\right) \qquad 0$
$\sum_{i=1}^n \mathrm{NegativeBinomial}(n_i,p) \sim \mathrm{NegativeBinomial}\left(\sum_{i=1}^n n_i,p\right) \qquad 0$
$\sum_{i=1}^n \mathrm{Geometric}(p) \sim \mathrm{NegativeBinomial}(n,p) \qquad 0$
$\sum_{i=1}^n \mathrm{Poisson}(\lambda_i) \sim \mathrm{Poisson}\left(\sum_{i=1}^n \lambda_i\right) \qquad \lambda_i>0$

Continuous distributions

$\sum_{i=1}^n \operatorname{Stable}\left(\alpha,\beta_i,c_i,\mu_i\right)=\operatorname{Stable}\left(\alpha,\frac{\sum_{i=1}^n \beta_i c_i ^\alpha}{\sum_{i=1}^n c_i^\alpha},\left( \sum_{i=1}^n c_i^\alpha \right)^{1/\alpha},\sum_{i=1}^n\mu_i\right)$

$\qquad 0<\alpha_i\le 2 \quad -1 \le \beta_i \le 1 \quad c_i>0 \quad \infty<\mu_i<\infty$

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

$\sum_{i=1}^n \operatorname{Normal}(\mu_i,\sigma_i^2) \sim \operatorname{Normal}\left(\sum_{i=1}^n \mu_i, \sum_{i=1}^n \sigma_i^2\right) \qquad -\infty<\mu_i<\infty \quad \sigma_i^2>0\quad (\alpha=2, \beta_i=0)$
$\sum_{i=1}^n \operatorname{Cauchy}(a_i,\gamma_i) \sim \operatorname{Cauchy}\left(\sum_{i=1}^n a_i, \sum_{i=1}^n \gamma_i\right) \qquad -\infty 0 \quad (\alpha=1, \beta_i=0)$
$\sum_{i=1}^n \operatorname{Levy}(\mu_i,c_i) \sim \operatorname{Levy}\left(\sum_{i=1}^n \mu_i, \left(\sum_{i=1}^n \sqrt{c_i}\right)^2\right) \qquad -\infty<\mu_i<\infty \quad c_i>0\quad (\alpha=1/2, \beta_i=1)$

$\sum_{i=1}^n \operatorname{Gamma}(\alpha_i,\beta) \sim \operatorname{Gamma}\left(\sum_{i=1}^n \alpha_i,\beta\right) \qquad \alpha_i>0 \quad \beta>0$
$\sum_{i=1}^n \operatorname{Voigt}(\mu_i,\gamma_i,\sigma_i) \sim \operatorname{Voigt}\left(\sum_{i=1}^n \mu_i,\sum_{i=1}^n \gamma_i,\sqrt{\sum_{i=1}^n \sigma_i^2}\right) \qquad -\infty<\mu_i<\infty \quad \gamma_i>0 \quad \sigma_i>0$ {{Cite web |year=2016 |orig-year=2012 |title=VoigtDistribution |url=https://reference.wolfram.com/language/ref/VoigtDistribution.html |access-date=2021-04-08 |website=Wolfram Language Documentation}}
$\sum_{i=1}^n \operatorname{VarianceGamma}(\mu_i,\alpha,\beta,\lambda_i) \sim \operatorname{VarianceGamma}\left(\sum_{i=1}^n \mu_i, \alpha,\beta, \sum_{i=1}^n \lambda_i\right) \qquad -\infty<\mu_i<\infty \quad \lambda_i > 0 \quad \sqrt{\alpha^2 - \beta^2} > 0$ {{Cite web |year=2012 |title=VarianceGammaDistribution |url=https://reference.wolfram.com/language/ref/VarianceGammaDistribution.html.en |access-date=2021-04-09 |website=Wolfram Language Documentation |publication-date=2016}}
$\sum_{i=1}^n \operatorname{Exponential}(\theta) \sim \operatorname{Erlang}(n,\theta) \qquad \theta>0 \quad n=1,2,\dots$
$\sum_{i=1}^n \operatorname{Exponential}(\lambda_i) \sim \operatorname{Hypoexponential}(\lambda_1,\dots,\lambda_n) \qquad \lambda_i>0$ {{Cite journal |arxiv=2012.08498 |first=George P. |last=Yanev |title=Exponential and Hypoexponential Distributions: Some Characterizations |journal=Mathematics |date=2020-12-15|volume=8 |issue=12 |page=2207 |doi=10.3390/math8122207 |doi-access=free }}
$\sum_{i=1}^n \chi^2(r_i) \sim \chi^2\left(\sum_{i=1}^n r_i\right) \qquad r_i=1,2,\dots$
$\sum_{i=1}^r N^2(0,1) \sim \chi^2_r \qquad r=1,2,\dots$
$\sum_{i=1}^n(X_i - \bar X)^2 \sim \sigma^2 \chi^2_{n-1}, \quad$ where $X_1,\dots,X_n$ is a random sample from $N(\mu,\sigma^2)$ and $\bar X = \frac{1}{n} \sum_{i=1}^n X_i.$

Mixed distributions:

$\operatorname{Normal}(\mu,\sigma^2)+\operatorname{Cauchy}(x_0,\gamma) \sim \operatorname{Voigt}(\mu+x_0,\sigma,\gamma)\qquad -\infty<\mu<\infty \quad -\infty 0 \quad \sigma>0$

References

Sources

{{cite book |last1=Hogg |first1=Robert V. |authorlink1=Robert V. Hogg |last2=McKean |first2=Joseph W. |last3=Craig |first3=Allen T. |title=Introduction to mathematical statistics |edition=6th |publisher=Prentice Hall |url=https://www.pearson.com/us/higher-education/product/Hogg-Introduction-to-Mathematical-Statistics-6th-Edition/9780130085078.html |location=Upper Saddle River, New Jersey |year=2004 | page=692 |isbn=978-0-13-008507-8 |mr=467974}}

Convolutions

Probability distributions, convolutions

list of convolutions of probability distributions

Discrete distributions

Continuous distributions

See also

References

Sources