List of convolutions of probability distributions

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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 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 and identically distributed random variables. 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<p<1 \quad n=1,2,\dots \,\!
  • \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 \,\!
  • \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 \,\!
  • \sum_{i=1}^n \mathrm{Geometric}(p) \sim \mathrm{NegativeBinomial}(n,p) \qquad 0<p<1 \quad n=1,2,\dots \,\!
  • \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 \mathrm{Normal}(\mu_i,\sigma_i^2) \sim \mathrm{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
  • \sum_{i=1}^n \mathrm{Cauchy}(a_i,\gamma_i) \sim \mathrm{Cauchy}\left(\sum_{i=1}^n a_i, \sum_{i=1}^n \gamma_i\right) \qquad -\infty<a_i<\infty \quad \gamma_i>0
  • \sum_{i=1}^n \mathrm{Gamma}(\alpha_i,\beta) \sim \mathrm{Gamma}\left(\sum_{i=1}^n \alpha_i,\beta\right) \qquad \alpha_i>0  \quad \beta>0
  • \sum_{i=1}^n \mathrm{Exponential}(\theta) \sim \mathrm{Gamma}(n,\theta) \qquad \theta>0 \quad n=1,2,\dots
  • \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. \,\!

See also


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