Reproductive property

The reproductive property of a probability distribution states that the sum of independent random variables of a certain distribution type is again distributed according to this type.

The normal distribution , the Poisson distribution , the gamma distribution , the chi-square distribution and the Cauchy distribution are reproductive . A property closely related to reproductivity is infinite divisibility .

example

The random variables and are independent and normally distributed as ${\displaystyle X_{1}}$ ${\displaystyle X_{2}}$

${\displaystyle X_{1}\sim {\mathcal {N}}(\mu _{1};\sigma _{1}^{2})\quad {\text{und}}\quad X_{2}\sim {\mathcal {N}}(\mu _{2},\sigma _{2}^{2})}$.

The random variable is then also normally distributed as ${\displaystyle Y=X_{1}+X_{2}}$

${\displaystyle Y\sim {\mathcal {N}}(\mu _{1}+\mu _{2},\sigma _{1}^{2}+\sigma _{2}^{2})}$.

The following generally applies: From independent it follows: ${\displaystyle X_{i}\sim {\mathcal {N}}(\mu _{i},\sigma _{i}^{2}),\quad i=1,\ldots ,k}$

${\displaystyle \sum \limits _{i=1}^{k}X_{i}\sim {\mathcal {N}}\left(\sum \limits _{i=1}^{k}\mu _{i},\sum \limits _{i=1}^{k}\sigma _{i}^{2}\right)}$.

Several parameters

If a distribution is described by two or more parameters, it can happen that there is only one parameter closed when the other parameters are retained. Are for example binomially distributed with parameters and , so and , so is . For the fixed the binomial distribution is reproductive with respect to . The above example of the normal distribution shows that seclusion can exist with several parameters even without such a restriction. ${\displaystyle X_{n},X_{m}}$ ${\displaystyle n,m}$ ${\displaystyle p}$ ${\displaystyle X_{n}\sim B_{n,p}}$ ${\displaystyle X_{m}\sim B_{m,p}}$ ${\displaystyle (X_{n}+X_{m})\sim B_{n+m,p}}$ ${\displaystyle p}$ ${\displaystyle B_{n,p}}$ ${\displaystyle n}$

Individual evidence

1. ^ Karl Mosler and Friedrich Schmid: Probability calculation and conclusive statistics. Springer-Verlag, 2011, p. 149.