List of univariate probability distributions

This list of univariate probability distributions gives an overview of the most popular univariate (one-dimensional) probability distributions .

Probability distributions describe how the probabilities are distributed among the possible outcomes of a random variable . A distinction is made between discrete distributions , which are defined on a finite or countable set, and continuous (continuous) distributions , which are usually defined on intervals.

Discrete distributions can be described by their counting density . For each of the maximum number of values ​​of a random variable that can be counted, this indicates the probability that exactly this value is obtained. ${\displaystyle x}$ ${\displaystyle X}$

In the case of continuous distributions , the probabilities of individual values ​​cannot be specified because they always have the probability . However, it is often possible to represent the probability that a random variable takes on a value in an interval as an integral over a density function (or probability density) : ${\displaystyle 0}$ ${\displaystyle X}$ ${\displaystyle [a,b]}$ ${\displaystyle f(x)}$

${\displaystyle P(a\leq X\leq b)=\int _{a}^{b}f(x)\,dx}$

With the continuous distributions included in this list, such a representation is possible via a density function.

Discrete distributions

The tables below summarize the parameters carrier , probability function , distribution function , expected value and variance of the following discrete distributions:

• Discrete equal distribution
• Bernoulli distribution (zero-one distribution)
• Binomial distribution
• Negative binomial distribution (Pascal distribution)
• Geometric distribution
• Hypergeometric distribution
• Poisson distribution
• Logarithmic distribution

It denotes the rounding function , the rounding function and a correspondingly distributed random variable. ${\displaystyle \lceil .\rceil }$ ${\displaystyle \lfloor .\rfloor }$ ${\displaystyle X}$

Discrete equal distribution

Range of values ​​of the parameters:	${\displaystyle n\in \mathbb {N} }$, ${\displaystyle k_{i}\in \mathbb {R} \;(i=1,\dots ,n)}$	Picture of the probability function: On , d. H. ${\displaystyle \{0,1,\dots ,20\}}$ ${\displaystyle n=21}$
Carrier:	${\displaystyle \{k_{i}:i=1,\dots ,n\}}$
Counting density:	${\displaystyle f(k_{i})={\frac {1}{n}}}$
Distribution function:	${\displaystyle P(\{X\leq x\})={\frac {|\{i:k_{i}\leq x\}|}{n}}}$
	${\displaystyle P(\{X<x\})={\frac {|\{i:k_{i}<x\}|}{n}}}$
Expected value:	${\displaystyle {\frac {1}{n}}\sum _{i=1}^{n}k_{i}}$
Variance:	${\displaystyle {\frac {1}{n}}\left(\sum _{i=1}^{n}k_{i}^{2}-{\frac {1}{n}}\left(\sum _{i=1}^{n}k_{i}\right)^{2}\right)}$

Bernoulli distribution (zero-one distribution)

Range of values ​​of the parameters:	${\displaystyle p\in [0,1]}$	Picture of the probability function: (blue), (green) and (red) ${\displaystyle p=0{,}2}$ ${\displaystyle p=0{,}5}$ ${\displaystyle p=0{,}8}$
Carrier:	${\displaystyle \{0,1\}}$
Counting density:	${\displaystyle f(k)={\begin{cases}p&{\text{für }}k=1\\1-p&{\text{für }}k=0\end{cases}}}$
Distribution function:	${\displaystyle P(\{X\leq x\})={\begin{cases}0&{\text{für }}x<0\\1-p&{\text{für }}0\leq x<1\\1&{\text{für }}x\geq 1\end{cases}}}$
	${\displaystyle P(\{X<x\})={\begin{cases}0&{\text{für }}x\leq 0\\1-p&{\text{für }}0<x\leq 1\\1&{\text{für }}x>1\end{cases}}}$
Expected value:	${\displaystyle p}$
Variance:	${\displaystyle p(1-p)}$

Binomial distribution

Range of values ​​of the parameters:	${\displaystyle n\in \mathbb {N} ^{+}}$, ${\displaystyle p\in [0,1]}$	Picture of the probability function: ; (blue), (green) and (red) ${\displaystyle n=20}$ ${\displaystyle p=0{,}1}$ ${\displaystyle p=0{,}5}$ ${\displaystyle p=0{,}8}$
Carrier:	${\displaystyle \{0,1,\dots ,n\}}$
Counting density:	${\displaystyle f(k)={n \choose k}p^{k}(1-p)^{n-k}}$
Distribution function:	${\displaystyle P(\{X\leq x\})=\sum _{i=0}^{\lfloor x\rfloor }{\binom {n}{i}}p^{i}(1-p)^{n-i}}$
	${\displaystyle P(\{X<x\})=\sum _{i=0}^{\lceil x-1\rceil }{n \choose i}p^{i}(1-p)^{n-i}}$
Expected value:	${\displaystyle np}$
Variance:	${\displaystyle np(1-p)}$

Negative binomial distribution (Pascal distribution)

Range of values ​​of the parameters:	${\displaystyle r\in \mathbb {N} ^{+}}$, ${\displaystyle p\in {]0,1]}}$	Picture of the probability function: ; (blue), (green) and (red) ${\displaystyle r=10}$ ${\displaystyle p=0{,}2}$ ${\displaystyle p=0{,}5}$ ${\displaystyle p=0{,}8}$
Carrier:	${\displaystyle \{x\in \mathbb {N} \colon x\geq r\}}$
Counting density:	${\displaystyle P(\{X=k\})={{k-1} \choose {r-1}}p^{r}(1-p)^{k-r}}$
Distribution function:	${\displaystyle P(\{X\leq x\})=\sum _{i=r}^{\lfloor x\rfloor }{i-1 \choose r-1}p^{r}(1-p)^{i-r}}$
	${\displaystyle P(\{X<x\})=\sum _{i=r}^{\lceil x-1\rceil }{i-1 \choose r-1}p^{r}(1-p)^{i-r}}$
Expected value:	${\displaystyle {\frac {r}{p}}}$
Variance:	${\displaystyle {\frac {r(1-p)}{p^{2}}}}$

Geometric distribution

option A

Range of values ​​of the parameters:	${\displaystyle p\in {]0,1[}}$	Picture of the probability function: (blue), (green) and (red) ${\displaystyle p=0{,}2}$ ${\displaystyle p=0{,}5}$ ${\displaystyle p=0{,}8}$
Carrier:	${\displaystyle \mathbb {N} ^{+}}$
Counting density:	${\displaystyle f(k)=p(1-p)^{k-1}}$
Distribution function:	${\displaystyle P(\{X\leq x\})=1-(1-p)^{\lfloor x\rfloor }}$
	${\displaystyle P(\{X<x\})=1-(1-p)^{\lceil x-1\rceil }}$
Expected value:	${\displaystyle {\frac {1}{p}}}$
Variance:	${\displaystyle {\frac {1}{p^{2}}}-{\frac {1}{p}}}$

Variant B

Range of values ​​of the parameters:	${\displaystyle p\in {]0,1[}}$	Picture of the probability function: (blue), (green) and (red) ${\displaystyle p=0{,}2}$ ${\displaystyle p=0{,}5}$ ${\displaystyle p=0{,}8}$
Carrier:	${\displaystyle \mathbb {N} _{0}}$
Counting density:	${\displaystyle f(k)=p(1-p)^{k}}$
Distribution function:	${\displaystyle P(\{X\leq x\})=1-(1-p)^{\lfloor x+1\rfloor }}$
	${\displaystyle P(\{X<x\})=1-(1-p)^{\lceil x\rceil }}$
Expected value:	${\displaystyle {\frac {1}{p}}-1}$
Variance:	${\displaystyle {\frac {1}{p^{2}}}-{\frac {1}{p}}}$

Hypergeometric distribution

Range of values ​​of the parameters:	${\displaystyle N\in \mathbb {N} ^{+}}$, with , with ${\displaystyle M\in \mathbb {N} ^{+}}$ ${\displaystyle M\leq N}$ ${\displaystyle n\in \mathbb {N} ^{+}}$ ${\displaystyle n\leq N}$	Picture of the probability function: ; (blue), (green) and (red) ${\displaystyle n=20}$ ${\displaystyle M=20,N=30}$ ${\displaystyle M=50,N=60}$ ${\displaystyle M=20,N=60}$
Carrier:	${\displaystyle \{\max(0,n+M-N),\dotsc ,\min(n,M)\}}$
Counting density:	${\displaystyle f(k)={\frac {\displaystyle {M \choose k}{N-M \choose n-k}}{\displaystyle {N \choose n}}}}$
Distribution function:	${\displaystyle P(\{X\leq x\})=\sum _{i=\max(0,n-N)}^{\lfloor x\rfloor }{\frac {{M \choose i}{N \choose n-i}}{M+N \choose n}}}$
	${\displaystyle P(\{X<x\})=\sum _{i=\max(0,n-N)}^{\lceil x-1\rceil }{\frac {{M \choose i}{N \choose n-i}}{M+N \choose n}}}$
Expected value:	${\displaystyle n{\frac {M}{N}}}$
Variance:	${\displaystyle n{\frac {M}{N}}\left(1-{\frac {M}{N}}\right){\frac {N-n}{N-1}}}$

Poisson distribution

Range of values ​​of the parameters:	${\displaystyle \lambda \in \mathbb {R} ^{+}}$	Picture of the probability function: (blue), (green) and (red) ${\displaystyle \lambda =1}$ ${\displaystyle \lambda =5}$ ${\displaystyle \lambda =10}$
Carrier:	${\displaystyle \mathbb {N} _{0}}$
Counting density:	${\displaystyle f(k)={\frac {\lambda ^{k}}{k!}}\cdot \mathrm {e} ^{-\lambda }}$
Distribution function:	${\displaystyle P(\{X\leq x\})=\sum _{i=0}^{\lfloor x\rfloor }{\frac {\lambda ^{i}}{i!}}\;\mathrm {e} ^{-\lambda }}$
	${\displaystyle P(\{X<x\})=\sum _{i=0}^{\lceil x-1\rceil }{\frac {\lambda ^{i}}{i!}}\;\mathrm {e} ^{-\lambda }}$
Expected value:	${\displaystyle \lambda }$
Variance:	${\displaystyle \lambda }$

Logarithmic distribution

Range of values ​​of the parameters:	${\displaystyle p\in (0,1)}$	Picture of the probability function: (blue), (green) and (red) ${\displaystyle p=0{,}2}$ ${\displaystyle p=0{,}5}$ ${\displaystyle p=0{,}8}$
Carrier:	${\displaystyle \mathbb {N} ^{+}}$
Counting density:	${\displaystyle f(k)={\frac {p^{k}}{k}}\cdot {\frac {1}{-\ln(1-p)}}}$
Distribution function:	${\displaystyle P(\{X\leq x\})=\sum _{i=0}^{\lfloor x\rfloor }{\frac {p^{i}}{i}}\cdot {\frac {1}{-\ln(1-p)}}}$
	${\displaystyle P(\{X<x\})=\sum _{i=0}^{\lceil x-1\rceil }{\frac {p^{i}}{i}}\cdot {\frac {1}{-\ln(1-p)}}}$
Expected value:	${\displaystyle {\frac {p}{-(1-p)\ln(1-p)}}}$
Variance:	${\displaystyle {\frac {p(-\ln(1-p)-p)}{(1-p)^{2}\ln ^{2}(1-p)}}}$

Continuous distributions

The tables below summarize the parameters carrier , density function , distribution function , expected value and variance of the following continuous distributions:

• Constant uniform distribution (rectangular distribution , uniform distribution)
• Triangular distribution (Simpson distribution)
• Normal distribution (Gaussian distribution)
• Logarithmic normal distribution
• Exponential distribution
• Chi-square distribution
• Student's t-distribution
• F distribution (Fisher distribution)
• Gamma distribution
• Beta distribution
• Logistic distribution
• Weibull distribution
• Cauchy distribution (Cauchy-Lorentz distribution, Lorentz distribution)
• Pareto distribution

The gamma function , the beta function and each denote a correspondingly distributed random variable with density and distribution function . ${\displaystyle \Gamma (r)}$ ${\displaystyle B(p,q)}$ ${\displaystyle X}$ ${\displaystyle f(x)}$ ${\displaystyle F(x)}$

Constant uniform distribution (rectangular distribution , uniform distribution)

Range of values ​​of the parameters:	${\displaystyle a,b\in \mathbb {R} }$ With ${\displaystyle a<b}$	Image of the density function: (blue), (green) and (red) ${\displaystyle a=4,b=8}$ ${\displaystyle a=1,b=18}$ ${\displaystyle a=1,b=11}$
Carrier:	${\displaystyle [a,b]}$
Density function:	${\displaystyle f(x)={\begin{cases}{\frac {1}{b-a}}&{\text{für }}a<x\leq b\\0&{\text{ sonst }}\end{cases}}}$
Distribution function:	${\displaystyle F(x)={\begin{cases}0&{\text{für }}x\leq a\\{\frac {x-a}{b-a}}&{\text{für }}a<x\leq b\\1&{\text{für }}x>b\end{cases}}}$
Expected value:	${\displaystyle {\frac {a+b}{2}}}$
Variance:	${\displaystyle {\frac {(b-a)^{2}}{12}}}$

Triangular distribution

Range of values ​​of the parameters:	${\displaystyle a,b,c\in \mathbb {R} }$ With ${\displaystyle a\leq c\leq b}$	Image of the density function:
Carrier:	${\displaystyle [a,b]}$
Density function:	${\displaystyle f(x)={\begin{cases}{\frac {2(x-a)}{(b-a)(c-a)}},&{\text{wenn }}a\leq x<c\\{\frac {2}{b-a}},&{\text{wenn }}x=c\\{\frac {2(b-x)}{(b-a)(b-c)}},&{\text{wenn }}c<x\leq b.\end{cases}}}$
Distribution function:	${\displaystyle F(x)={\begin{cases}{\frac {(x-a)^{2}}{(b-a)(c-a)}},&{\text{wenn }}a\leq x<c\\{\frac {c-a}{b-a}},&{\text{wenn }}x=c\\1-{\frac {(b-x)^{2}}{(b-a)(b-c)}},&{\text{wenn }}c<x\leq b.\end{cases}}}$
Expected value:	${\displaystyle \operatorname {E} (X)={\frac {a+b+c}{3}}.}$
Variance:	${\displaystyle \operatorname {Var} (X)={\frac {(a-b)^{2}+(b-c)^{2}+(a-c)^{2}}{36}}.}$

Normal distribution (Gaussian distribution)

Range of values ​​of the parameters:	${\displaystyle \mu \in \mathbb {R} }$ and ${\displaystyle \sigma \in \mathbb {R} ^{+}}$	Image of the density function: (blue), (green) and (red) ${\displaystyle \mu =0,\sigma =1}$ ${\displaystyle \mu =0,\sigma =2}$ ${\displaystyle \mu =-1,\sigma =2}$
Carrier:	${\displaystyle \mathbb {R} }$
Density function:	${\displaystyle f(x)={\frac {1}{\sigma {\sqrt {2\pi }}}}\,\mathrm {e} ^{-{\frac {1}{2}}\left({\frac {x-\mu }{\sigma }}\right)^{2}}}$
Distribution function:	${\displaystyle F(x)={\frac {1}{\sigma {\sqrt {2\pi }}}}\cdot \int _{-\infty }^{x}\mathrm {e} ^{-{\frac {1}{2}}\cdot \left({\frac {t-\mu }{\sigma }}\right)^{2}}\mathrm {d} t}$
Expected value:	${\displaystyle \mu }$
Variance:	${\displaystyle \sigma ^{2}}$

Log normal distribution (log normal distribution)

Range of values ​​of the parameters:	${\displaystyle \mu \in \mathbb {R} }$ and ${\displaystyle \sigma \in \mathbb {R} ^{+}}$	Image of the density function: (blue), (green) and (red) ${\displaystyle \mu =0,\sigma =1}$ ${\displaystyle \mu =0,\sigma =2}$ ${\displaystyle \mu =-1,\sigma =2}$
Carrier:	${\displaystyle \mathbb {R} _{0}^{+}}$
Density function:	${\displaystyle f(x)={\frac {1}{\sigma {\sqrt {2\pi }}}}\,{\frac {1}{x}}\,\mathrm {e} ^{-{\frac {1}{2}}\left({\frac {\operatorname {ln} \,x-\mu }{\sigma }}\right)^{2}}}$
Distribution function:	${\displaystyle F(x)={\begin{cases}0&{\text{für }}x\leq 0\\{\frac {1}{\sigma \cdot {\sqrt {2\pi }}}}\cdot \int _{0}^{x}\,{\frac {1}{t}}\,\mathrm {e} ^{-{\frac {1}{2}}\cdot \left({\frac {\operatorname {ln} \,t-\mu }{\sigma }}\right)^{2}}\mathrm {d} t&{\text{für }}x>0\end{cases}}}$
Expected value:	${\displaystyle \exp(\mu +\sigma ^{2}/2)}$
Variance:	${\displaystyle \exp(2\mu +\sigma ^{2})\cdot (\exp(\sigma ^{2})-1)}$

Exponential distribution

Range of values ​​of the parameters:	${\displaystyle \alpha \in \mathbb {R} ^{+}}$	Image of the density function: (blue), (green) and (red) ${\displaystyle \alpha =1}$ ${\displaystyle \alpha =5}$ ${\displaystyle \alpha =10}$
Carrier:	${\displaystyle \mathbb {R} _{0}^{+}}$
Density function:	${\displaystyle f(x)=\alpha \cdot \mathrm {e} ^{-\alpha x}}$
Distribution function:	${\displaystyle F(x)={\begin{cases}0&{\text{für }}x\leq 0\\1-\mathrm {e} ^{-\alpha x}&{\text{für }}x>0\end{cases}}}$
Expected value:	${\displaystyle {\frac {1}{\alpha }}}$
Variance:	${\displaystyle {\frac {1}{\alpha ^{2}}}}$

Chi-square distribution

Range of values ​​of the parameters:	${\displaystyle n\in \mathbb {N} ^{+}}$	Image of the density function: (blue), (green) and (red) ${\displaystyle n=2}$ ${\displaystyle n=5}$ ${\displaystyle n=10}$
Carrier:	${\displaystyle \mathbb {R} _{0}^{+}}$
Density function:	${\displaystyle f_{n}(x)={\frac {1}{2^{\frac {n}{2}}\Gamma ({\tfrac {n}{2}})}}x^{{\frac {n}{2}}-1}\operatorname {exp} \left\{-{\frac {x}{2}}\right\}}$
Distribution function:	${\displaystyle F(x)={\begin{cases}\displaystyle 0&{\text{für }}x\leq 0\\1-{\frac {\Gamma \left({\frac {n}{2}},{\frac {x}{2}}\right)}{\Gamma \left({\frac {n}{2}}\right)}}&{\text{für }}x>0\end{cases}}}$
Expected value:	${\displaystyle n}$
Variance:	${\displaystyle 2n}$

Student's t-distribution

Range of values ​​of the parameters:	${\displaystyle k\in \mathbb {N} ^{+}}$	Image of the density function: (blue), (green) and (red) ${\displaystyle k=2}$ ${\displaystyle k=5}$ ${\displaystyle k=10}$
Carrier:	${\displaystyle \mathbb {R} }$
Density function:	${\displaystyle f(x)={\frac {\Gamma ({\frac {k+1}{2}})}{\Gamma ({\frac {k}{2}})\,{\sqrt {k\,\pi \,}}}}\,\cdot \,\left(1+{\frac {x^{2}}{k}}\right)^{-{\frac {k+1}{2}}}}$
Distribution function:	${\displaystyle F(x)={\frac {\Gamma ({\frac {k+1}{2}})}{\Gamma ({\frac {k}{2}})\,{\sqrt {k\,\pi \,}}}}\,\cdot \,\int _{-\infty }^{x}\,\left(1+{\frac {t^{2}}{k}}\right)^{-{\frac {k+1}{2}}}\mathrm {d} t}$
Expected value:	${\displaystyle 0}$
Variance:	${\displaystyle {\frac {k}{k-2}}}$

F distribution (Fisher distribution)

Range of values ​​of the parameters:	${\displaystyle m\in \mathbb {N} ^{+}}$ and ${\displaystyle n\in \mathbb {N} ^{+}}$	Image of the density function: (blue), (green) and (red) ${\displaystyle m=2,n=10}$ ${\displaystyle m=10,n=10}$ ${\displaystyle m=10,n=2}$
Carrier:	${\displaystyle \mathbb {R} _{0}^{+}}$
Density function:	${\displaystyle f(x)={\frac {\Gamma ({\frac {m+n}{2}})\,\left({\frac {m}{n}}\right)^{\frac {m}{2}}}{\Gamma ({\frac {m}{2}})\,\Gamma ({\frac {n}{2}})}}x^{({\frac {m}{2}}-1)}\left(1+{\frac {m}{n}}x\right)^{(-{\frac {m+n}{2}})}}$
Distribution function:	${\displaystyle F(x)={\begin{cases}0\\\quad {\text{für }}x\leq 0\\{\frac {\Gamma ({\frac {m+n}{2}})\,\left({\frac {m}{n}}\right)^{\frac {m}{2}}}{\Gamma ({\frac {m}{2}})\,\Gamma ({\frac {n}{2}})}}\int _{0}^{x}\,t^{({\frac {m}{2}}-1)}\left(1+{\frac {m}{n}}t\right)^{(-{\frac {m+n}{2}})}\mathrm {d} t\\\quad {\text{für }}x>0\end{cases}}}$
Expected value:	${\displaystyle {\frac {n}{n-2}}}$(only defined for ) ${\displaystyle n>2}$
Variance:	${\displaystyle {\frac {2n^{2}(m+n-2)}{m(n-2)^{2}(n-4)}}}$(only defined for ) ${\displaystyle n>4}$

Gamma distribution

Range of values ​​of the parameters:	${\displaystyle p\in \mathbb {R} ^{+}}$ and ${\displaystyle b\in \mathbb {R} ^{+}}$	Image of the density function: (blue), (green) and (red) ${\displaystyle p=0{,}5,b=2}$ ${\displaystyle p=1,b=1}$ ${\displaystyle p=2,b=1}$
Carrier:	${\displaystyle \mathbb {R} _{0}^{+}}$
Density function:	${\displaystyle f(x)={b^{p} \over \Gamma (p)}x^{p-1}\mathrm {e} ^{-bx}}$
Distribution function:	${\displaystyle F(x)={\begin{cases}0&{\text{für }}x\leq 0\\{b^{p} \over \Gamma (p)}\,\cdot \,\int _{0}^{x}\,t^{p-1}\mathrm {e} ^{-bt}\mathrm {d} t&{\text{für }}x>0\end{cases}}}$
Expected value:	${\displaystyle {\frac {p}{b}}}$
Variance:	${\displaystyle {\frac {p}{b^{2}}}}$

Beta distribution

Range of values ​​of the parameters:	${\displaystyle p\in \mathbb {R} ^{+}}$ and ${\displaystyle q\in \mathbb {R} ^{+}}$	Image of the density function: (blue), (green) and (red) ${\displaystyle p=0{,}5,q=2}$ ${\displaystyle p=2,q=2}$ ${\displaystyle p=2,q=5}$
Carrier:	${\displaystyle [0,1]}$
Density function:	${\displaystyle f(x)={\frac {1}{B(p,q)}}x^{p-1}(1-x)^{q-1}}$
Distribution function:	${\displaystyle F(x)={\begin{cases}0&{\text{für }}x<0\\{{1} \over {B(p,q)}}\int _{0}^{x}u^{p-1}(1-u)^{q-1}\mathrm {d} u&{\text{für }}0\leq x\leq 1\\1&{\text{für }}x>1\end{cases}}}$
Expected value:	${\displaystyle {\frac {p}{p+q}}}$
Variance:	${\displaystyle {\frac {pq}{(p+q+1)(p+q)^{2}}}}$

Logistic distribution

Range of values ​​of the parameters:	${\displaystyle \alpha \in \mathbb {R} }$ and ${\displaystyle \beta \in \mathbb {R} ^{+}}$	Image of the density function: (blue), (green) and (red) ${\displaystyle \alpha =0,\beta =1}$ ${\displaystyle \alpha =0,\beta =2}$ ${\displaystyle \alpha =-1,\beta =1}$
Carrier:	${\displaystyle \mathbb {R} }$
Density function:	${\displaystyle f(x)={\frac {\mathrm {e} ^{-{\frac {x-\alpha }{\beta }}}}{\beta \left(1+\mathrm {e} ^{-{\frac {x-\alpha }{\beta }}}\right)^{2}}}}$
Distribution function:	${\displaystyle F(x)={\frac {1}{1+\mathrm {e} ^{-{\frac {x-\alpha }{\beta }}}}}}$
Expected value:	${\displaystyle \alpha }$
Variance:	${\displaystyle {\frac {\beta ^{2}\pi ^{2}}{3}}}$

Weibull distribution

Range of values ​​of the parameters:	${\displaystyle \alpha \in \mathbb {R} ^{+}}$ and ${\displaystyle \beta \in \mathbb {R} ^{+}}$	Image of the density function: (blue), (green) and (red) ${\displaystyle \alpha =1,\beta =1}$ ${\displaystyle \alpha =1,\beta =2}$ ${\displaystyle \alpha =5,\beta =3}$
Carrier:	${\displaystyle \mathbb {R} _{0}^{+}}$
Density function:	${\displaystyle f(x)=\alpha \beta x^{\beta -1}\mathrm {e} ^{-\alpha x^{\beta }}}$
Distribution function:	${\displaystyle F(x)={\begin{cases}1-\mathrm {e} ^{-\alpha x^{\beta }}&{\text{für }}x>0\\0&{\text{für }}x\leq 0\end{cases}}}$
Expected value:	${\displaystyle \alpha ^{-1/\beta }\cdot \Gamma \left({\frac {1}{\beta }}+1\right)}$
Variance:	${\displaystyle \alpha ^{-2/\beta }\cdot \left(\Gamma \left({\frac {2}{\beta }}+1\right)-\Gamma \left({\frac {1}{\beta }}+1\right)^{2}\right)}$

Cauchy distribution (Cauchy-Lorentz distribution, Lorentz distribution)

Range of values ​​of the parameters:	${\displaystyle s\in \mathbb {R} ^{+}}$ and ${\displaystyle t\in \mathbb {R} }$	Image of the density function: (blue), (green) and (red) ${\displaystyle s=1,t=0}$ ${\displaystyle s=2,t=0}$ ${\displaystyle s=2,t=-1}$
Carrier:	${\displaystyle \mathbb {R} }$
Density function:	${\displaystyle f(x)={\frac {1}{\pi }}\cdot {\frac {s}{s^{2}+(x-t)^{2}}}}$
Distribution function:	${\displaystyle F(x)={\frac {1}{2}}+{\frac {1}{\pi }}\cdot \arctan \left({\frac {x-t}{s}}\right)}$
Expected value:	not defined
Variance:	not defined

Pareto distribution

Range of values ​​of the parameters:	${\displaystyle x_{\min }\in \mathbb {R} ^{+}}$ and ${\displaystyle k\in \mathbb {R} ^{+}}$	Image of the density function: (blue), (green) and (red) ${\displaystyle x_{\min }=1,k=1}$ ${\displaystyle x_{\min }=1,k=2}$ ${\displaystyle x_{\min }=2,k=1}$
Carrier:	${\displaystyle [x_{\min },\infty )}$
Density function:	${\displaystyle f(x)={\frac {k}{x_{\min }}}\left({\frac {x_{\min }}{x}}\right)^{k+1}}$
Distribution function:	${\displaystyle F(x)=1-\left({\frac {x_{\min }}{x}}\right)^{k}}$
Expected value:	${\displaystyle {\begin{cases}\displaystyle x_{\min }{\frac {k}{k-1}}&{\text{für }}k>1\\\infty &{\text{für }}k\leq 1\end{cases}}}$
Variance:	${\displaystyle {\begin{cases}\displaystyle x_{\min }^{2}{\frac {k}{(k-2)(k-1)^{2}}}&{\text{für }}k>2\\\infty &{\text{für }}k\leq 2\end{cases}}}$

See also

• List of relationships between probability distributions
• List of multivariate and matrix-variable probability distributions

Web links

• Interactive illustrations of distributions

This version was added to the selection of informative lists and portals on December 12, 2009 .