Name | Distribution |
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Arcsine distribution | F ( x ) = 2 π arcsin ( x ) = arcsin ( 2 x − 1 ) π + 1 2 {\displaystyle F(x)={\frac {2}{\pi }}\arcsin \left({\sqrt {x}}\right)={\frac {\arcsin(2x-1)}{\pi }}+{\frac {1}{2}}} for 0 ≤ x ≤ 1 f ( x ) = 1 π x ( 1 − x ) {\displaystyle f(x)={\frac {1}{\pi {\sqrt {x(1-x)}}}}} on (0,1) |
Bates distribution | f X ( x ; n ) = n 2 ( n − 1 ) ! ∑ k = 0 n ( − 1 ) k ( n k ) ( n x − k ) n − 1 sgn ( n x − k ) {\displaystyle f_{X}(x;n)={\frac {n}{2(n-1)!}}\sum _{k=0}^{n}(-1)^{k}{n \choose k}(nx-k)^{n-1}\operatorname {sgn}(nx-k)} |
Cauchy distribution | f ( x ; x 0 , γ ) = 1 π γ [ 1 + ( x − x 0 γ ) 2 ] = 1 π γ [ γ 2 ( x − x 0 ) 2 + γ 2 ] , {\displaystyle f(x;x_{0},\gamma )={\frac {1}{\pi \gamma \left[1+\left({\frac {x-x_{0}}{\gamma }}\right)^{2}\right]}}={1 \over \pi \gamma }\left[{\gamma ^{2} \over (x-x_{0})^{2}+\gamma ^{2}}\right],} |
Champernowne distribution | f ( y ; α , λ , y 0 ) = n cosh [ α ( y − y 0 ) ] + λ , − ∞ < y < ∞ , {\displaystyle f(y;\alpha ,\lambda ,y_{0})={\frac {n}{\cosh[\alpha (y-y_{0})]+\lambda }},\qquad -\infty <y<\infty ,} |
Continuous uniform distribution | f ( x ) = { 1 b − a f o r a ≤ x ≤ b , 0 f o r x < a o r x > b {\displaystyle f(x)={\begin{cases}{\frac {1}{b-a}}&\mathrm {for} \ a\leq x\leq b,\\[8pt]0&\mathrm {for} \ x<a\ \mathrm {or} \ x>b\end{cases}}} |
Degenerate distribution | F k 0 ( x ) = { 1 , if x ≥ k 0 0 , if x < k 0 {\displaystyle F_{k_{0}}(x)=\left\{{\begin{matrix}1,&{\mbox{if }}x\geq k_{0}\\0,&{\mbox{if }}x<k_{0}\end{matrix}}\right.} |
Discrete uniform distribution | F ( k ; a , b ) = ⌊ k ⌋ − a + 1 b − a + 1 {\displaystyle F(k;a,b)={\frac {\lfloor k\rfloor -a+1}{b-a+1}}} |
Elliptical distribution | f ( x ) = k ⋅ g ( ( x − μ ) ′ Σ − 1 ( x − μ ) ) {\displaystyle f(x)=k\cdot g((x-\mu )'\Sigma ^{-1}(x-\mu ))} |
Gaussian q-distribution | s q ( x ) = { 0 if x < − ν 1 c ( q ) E q 2 − q 2 x 2 [ 2 ] q if − ν ≤ x ≤ ν 0 if x > ν . {\displaystyle s_{q}(x)={\begin{cases}0&{\text{if }}x<-\nu \\{\frac {1}{c(q)}}E_{q^{2}}^{\frac {-q^{2}x^{2}}{[2]_{q}}}&{\text{if }}-\nu \leq x\leq \nu \\0&{\mbox{if }}x>\nu .\end{cases}}} |
Hyperbolic distribution with asymmetry parameter equal to zero | γ 2 α δ K 1 ( δ γ ) e − α δ 2 + ( x − μ ) 2 + β ( x − μ ) {\displaystyle {\frac {\gamma }{2\alpha \delta K_{1}(\delta \gamma )}}\;e^{-\alpha {\sqrt {\delta ^{2}+(x-\mu )^{2}}}+\beta (x-\mu )}} K λ {\displaystyle K_{\lambda }} denotes a modified Bessel function of the second kind |
Generalized normal distribution | β 2 α Γ ( 1 / β ) e − ( | x − μ | / α ) β {\displaystyle {\frac {\beta }{2\alpha \Gamma (1/\beta )}}\;e^{-(|x-\mu |/\alpha )^{\beta }}} Γ {\displaystyle \Gamma } denotes the gamma function |
Hyperbolic secant distribution | f ( x ) = 1 2 sech ( π 2 x ) , {\displaystyle f(x)={\frac {1}{2}}\;\operatorname {sech} \!\left({\frac {\pi }{2}}\,x\right)\!,} |
Laplace distribution | f ( x ∣ μ , b ) = 1 2 b exp ( − | x − μ | b ) {\displaystyle f(x\mid \mu ,b)={\frac {1}{2b}}\exp \left(-{\frac {|x-\mu |}{b}}\right)\,\!} = 1 2 b { exp ( − μ − x b ) if x < μ exp ( − x − μ b ) if x ≥ μ {\displaystyle ={\frac {1}{2b}}\left\{{\begin{matrix}\exp \left(-{\frac {\mu -x}{b}}\right)&{\text{if }}x<\mu \\[8pt]\exp \left(-{\frac {x-\mu }{b}}\right)&{\text{if }}x\geq \mu \end{matrix}}\right.} |
Irwin-Hall distribution | f X ( x ; n ) = 1 2 ( n − 1 ) ! ∑ k = 0 n ( − 1 ) k ( n k ) ( x − k ) n − 1 sgn ( x − k ) {\displaystyle f_{X}(x;n)={\frac {1}{2(n-1)!}}\sum _{k=0}^{n}(-1)^{k}{n \choose k}(x-k)^{n-1}\operatorname {sgn}(x-k)} |
Logistic distribution | f ( x ; 0 , 1 ) = e − x ( 1 + e − x ) 2 = 1 ( e x / 2 + e − x / 2 ) 2 = 1 4 sech 2 ( x 2 ) . {\displaystyle {\begin{aligned}f(x;0,1)&={\frac {e^{-x}}{(1+e^{-x})^{2}}}\\[4pt]&={\frac {1}{(e^{x/2}+e^{-x/2})^{2}}}\\[5pt]&={\frac {1}{4}}\operatorname {sech} ^{2}\left({\frac {x}{2}}\right).\end{aligned}}} |
Normal distribution | φ ( x ) = e − x 2 2 2 π {\displaystyle \varphi (x)={\frac {e^{-{\frac {x^{2}}{2}}}}{\sqrt {2\pi }}}} |
Normal-exponential-gamma distribution | f ( x ; μ , k , θ ) ∝ exp ( ( x − μ ) 2 4 θ 2 ) D − 2 k − 1 ( | x − μ | θ ) {\displaystyle f(x;\mu ,k,\theta )\propto \exp {\left({\frac {(x-\mu )^{2}}{4\theta ^{2}}}\right)}D_{-2k-1}\left({\frac {|x-\mu |}{\theta }}\right)} |
Rademacher distribution | f ( k ) = { 1 / 2 if k = − 1 , 1 / 2 if k = + 1 , 0 otherwise. {\displaystyle f(k)=\left\{{\begin{matrix}1/2&{\mbox{if }}k=-1,\\1/2&{\mbox{if }}k=+1,\\0&{\mbox{otherwise.}}\end{matrix}}\right.} |
Raised cosine distribution | f ( x ; μ , s ) = 1 2 s [ 1 + cos ( x − μ s π ) ] = 1 s hvc ( x − μ s π ) {\displaystyle f(x;\mu ,s)={\frac {1}{2s}}\left[1+\cos \left({\frac {x-\mu }{s}}\,\pi \right)\right]\,={\frac {1}{s}}\operatorname {hvc} \left({\frac {x-\mu }{s}}\,\pi \right)\,} |
Student's distribution | f ( t ) = Γ ( ν + 1 2 ) ν π Γ ( ν 2 ) ( 1 + t 2 ν ) − ν + 1 2 , {\displaystyle f(t)={\frac {\Gamma ({\frac {\nu +1}{2}})}{{\sqrt {\nu \pi }}\,\Gamma ({\frac {\nu }{2}})}}\left(1+{\frac {t^{2}}{\nu }}\right)^{\!-{\frac {\nu +1}{2}}},\!} |
U-quadratic distribution | f ( x | a , b , α , β ) = α ( x − β ) 2 , for x ∈ [ a , b ] . {\displaystyle f(x|a,b,\alpha ,\beta )=\alpha \left(x-\beta \right)^{2},\quad {\text{for }}x\in [a,b].} |
Voigt distribution | V ( x ; σ , γ ) ≡ ∫ − ∞ ∞ G ( x ′ ; σ ) L ( x − x ′ ; γ ) d x ′ , {\displaystyle V(x;\sigma ,\gamma )\equiv \int _{-\infty }^{\infty }G(x';\sigma )L(x-x';\gamma )\,dx',} |
von Mises distribution | f ( x ∣ μ , κ ) = e κ cos ( x − μ ) 2 π I 0 ( κ ) {\displaystyle f(x\mid \mu ,\kappa )={\frac {e^{\kappa \cos(x-\mu )}}{2\pi I_{0}(\kappa )}}} |
Wigner semicircle distribution | f ( x ) = 2 π R 2 R 2 − x 2 {\displaystyle f(x)={2 \over \pi R^{2}}{\sqrt {R^{2}-x^{2}\,}}\,} |