List of relationships between probability distributions

The following table provides an overview of the relationships between probability distributions .

description	Memory aid *)
The square of a standard normally distributed random variable is chi-square distributed with parameter 1.	${\displaystyle N_{0,1}^{2}=C_{1}}$
The sum of independent chi-square distributed random variables is again chi-square distributed.	${\displaystyle C_{k}+C_{l}=C_{k+l}}$
The sum of independent normally distributed random variables is again normally distributed.	${\displaystyle N_{\mu _{1},\sigma _{1}^{2}}+N_{\mu _{2},\sigma _{2}^{2}}=N_{\mu _{1}+\mu _{2},\sigma _{1}^{2}+\sigma _{2}^{2}}}$
The sum of independent Poisson-distributed random variables is again Poisson-distributed.	${\displaystyle P_{\alpha }+P_{\beta }=P_{\alpha +\beta }}$
The sum of independent binomially distributed random variables with the same parameter p is again binomially distributed.	${\displaystyle B_{m,p}+B_{n,p}=B_{m+n,p}}$
The sum of independent, negatively binomially distributed random variables with the same parameter p is again negatively binomially distributed.	${\displaystyle NB_{m,p}+NB_{n,p}=NB_{m+n,p}}$
The sum of independent Erlang-distributed random variables with the same parameter is again Erlang-distributed. ${\displaystyle \alpha }$	${\displaystyle E_{\alpha ,m}+E_{\alpha ,n}=E_{\alpha ,m+n}}$
The sum of independent gamma-distributed random variables with the same parameter b is again gamma-distributed.	${\displaystyle G_{b,p1}+G_{b,p2}=G_{b,(p1+p2)}}$
Relationship between the Erlang distribution and the exponential distribution .	${\displaystyle E_{\alpha ,1}=E_{\alpha }}$
Relationship between the Erlang distribution and the chi-square distribution.	${\displaystyle E_{{\frac {1}{2}},n}=C_{2n}}$
Relationship between Erlangen distribution and gamma distribution. (For the second integer parameter, the gamma distribution agrees with the Erlang distribution.)	${\displaystyle E_{\alpha ,n}=G_{\alpha ,n}}$
Relationship between the Weibull distribution and the exponential distribution.	${\displaystyle W_{0,\sigma ,1}=E_{\frac {1}{\sigma }}}$
If X and Y are independent random variables, X are standard normal distributed and Y are -distributed, then -distributed. ${\displaystyle C_{k}}$ ${\displaystyle {\frac {X}{\sqrt {\frac {Y}{k}}}}}$ ${\displaystyle T_{k}}$	${\displaystyle {\frac {N_{0,1}}{\sqrt {\frac {C_{k}}{k}}}}=T_{k}}$
If X and Y are independent random variables, X -distributed and Y -distributed, then -distributed. ${\displaystyle C_{m}}$ ${\displaystyle C_{n}}$ ${\displaystyle {\frac {\frac {X}{m}}{\frac {Y}{n}}}}$ ${\displaystyle F_{m,n}}$	${\displaystyle {\frac {\frac {C_{m}}{m}}{\frac {C_{n}}{n}}}=F_{m,n}}$
The logarithm of a logarithmically normally distributed random variable is normally distributed.	${\displaystyle \ln(LN_{\mu ,\sigma ^{2}})=N_{\mu ,\sigma ^{2}}}$
If Z is negatively binomially distributed with parameters 1 and p , then Z - 1 is geometrically distributed with parameter p .	${\displaystyle NB_{1,p}-1=G_{p}}$

*) In the memory aid, for example, it does not stand for the chi-square distribution , but for a random variable in a chi-square distribution. The difference is that, for example, the distribution of the sum of random variables (it is called the convolution of the distributions) is usually written with, for example, ( distributions) instead of ( random variable) as is the case here . The advantage of the notation ( distributions) is that it already indicates which operation is to be applied to the distribution functions in order to obtain the distribution of the sum. The advantage of the notation ( random variable) is that it indicates which operation originally acted on the random variable. ${\displaystyle C_{k}}$ ${\displaystyle C_{k}*C_{l}}$ ${\displaystyle C_{k},C_{l}}$ ${\displaystyle C_{k}+C_{l}}$ ${\displaystyle C_{k},C_{l}}$ ${\displaystyle C_{k}*C_{l}}$ ${\displaystyle C_{k},C_{l}}$ ${\displaystyle C_{k}+C_{l}}$ ${\displaystyle C_{k},C_{l}}$

The "=" sign stands for "has the same distribution as".

The random variables that are on the left side of the equal sign are always completely independent of one another.

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From the above rules as follows (in "memory aid" notation) . Note that the first random variable must be independent of the second random variable . If instead you use the same random variable both times, i.e. when you calculate, the result is different! ${\displaystyle N_{0,1}^{2}+N_{0,1}^{2}=C_{1}+C_{1}=C_{2}=E_{{\frac {1}{2}},1}=E_{\frac {1}{2}}}$ ${\displaystyle N_{0,1}\,}$ ${\displaystyle N_{0,1}\,}$ ${\displaystyle 2\cdot N_{0,1}^{2}}$

See also

• List of univariate probability distributions

Web links

• Overview of other important relationships of probability distributions (PDF; 319 kB)