Reproductivity

We say a distribution is reproductive if the sum of two Random Variables of this distribution is still in this distribution. The easiest way to prove reproductivity is to use MGF or CF.

• Reproductive distributions
  • Poisson Distribution
    • Parameter: ${\lambda }_1,{\lambda }_2\to {\lambda }_1+{\lambda }_2$
  • Normal Distribution
    • Parameter: $({\mu }_1,{\sigma }_1^2),({\mu }_2,{\sigma }_2^2)\to ({\mu }_1+{\mu }_2,{\sigma }_1^2+{\sigma }_2^2)$
  • Chi-Square Distribution
    • Parameter: $n_1,n_2\to n_1+n_2$
• Reproductive under $\min$ operator
  • Exponential Distribution
    • Parameter: ${\lambda }_1,{\lambda }_2\to {\lambda }_1+{\lambda }_2$
  • Geometric Distribution
    • Parameter: $q_1,q_2\to q_1{q}_2$
• Reproductive when one of the parameters is the same
  • Binomial Distribution
    • Parameters: $(n_1,p),(n_2,p)\to (n_1+n_2,p)$
  • Gamma Distribution
    • Parameters: $({\alpha }_1,\lambda ),({\alpha }_2,\lambda )\to ({\alpha }_1+{\alpha }_2,\lambda )$
• Sample variance is of the same distribution
  • Cauchy Distribution