Continuous probability distribution
The Kaniadakis Erlang distribution (or κ-Erlang Gamma distribution ) is a family of continuous statistical distributions , which is a particular case of the κ-Gamma distribution , when α = 1 {\displaystyle \alpha =1} ν = n = {\displaystyle \nu =n=} κ-exponential distribution of Type I. The κ-Erlang is a κ-deformed version of the Erlang distribution . It is one example of a Kaniadakis distribution .
Characterization
Probability density function The Kaniadakis κ -Erlang distribution has the following probability density function :
f κ ( x ) = 1 ( n − 1 ) ! ∏ m = 0 n [ 1 + ( 2 m − n ) κ ] x n − 1 exp κ ( − x ) {\displaystyle f_{_{\kappa }}(x)={\frac {1}{(n-1)!}}\prod _{m=0}^{n}\left[1+(2m-n)\kappa \right]x^{n-1}\exp _{\kappa }(-x)} valid for x ≥ 0 {\displaystyle x\geq 0} n = positive integer {\displaystyle n={\textrm {positive}}\,\,{\textrm {integer}}} 0 ≤ | κ | < 1 {\displaystyle 0\leq |\kappa |<1} Kaniadakis entropy .
The ordinary Erlang Distribution is recovered as κ → 0 {\displaystyle \kappa \rightarrow 0}
Cumulative distribution function The cumulative distribution function of κ -Erlang distribution assumes the form:
F κ ( x ) = 1 ( n − 1 ) ! ∏ m = 0 n [ 1 + ( 2 m − n ) κ ] ∫ 0 x z n − 1 exp κ ( − z ) d z {\displaystyle F_{\kappa }(x)={\frac {1}{(n-1)!}}\prod _{m=0}^{n}\left[1+(2m-n)\kappa \right]\int _{0}^{x}z^{n-1}\exp _{\kappa }(-z)dz} valid for x ≥ 0 {\displaystyle x\geq 0} 0 ≤ | κ | < 1 {\displaystyle 0\leq |\kappa |<1} Erlang distribution is recovered in the classical limit κ → 0 {\displaystyle \kappa \rightarrow 0}
Survival distribution and hazard functions The survival function of the κ -Erlang distribution is given by:
The survival function of the κ -Erlang distribution enables the determination of hazard functions in closed form through the solution of the κ -rate equation:
where h κ {\displaystyle h_{\kappa }}
Family distribution A family of κ -distributions arises from the κ -Erlang distribution, each associated with a specific value of n {\displaystyle n} x ≥ 0 {\displaystyle x\geq 0} 0 ≤ | κ | < 1 {\displaystyle 0\leq |\kappa |<1} κ -Erlang cumulative distribution, which can be rewritten as:
F κ ( x ) = 1 − [ R κ ( x ) + Q κ ( x ) 1 + κ 2 x 2 ] exp κ ( − x ) {\displaystyle F_{\kappa }(x)=1-\left[R_{\kappa }(x)+Q_{\kappa }(x){\sqrt {1+\kappa ^{2}x^{2}}}\right]\exp _{\kappa }(-x)} where
Q κ ( x ) = N κ ∑ m = 0 n − 3 ( m + 1 ) c m + 1 x m + N κ 1 − n 2 κ 2 x n − 1 {\displaystyle Q_{\kappa }(x)=N_{\kappa }\sum _{m=0}^{n-3}\left(m+1\right)c_{m+1}x^{m}+{\frac {N_{\kappa }}{1-n^{2}\kappa ^{2}}}x^{n-1}} R κ ( x ) = N κ ∑ m = 0 n c m x m {\displaystyle R_{\kappa }(x)=N_{\kappa }\sum _{m=0}^{n}c_{m}x^{m}} with
N κ = 1 ( n − 1 ) ! ∏ m = 0 n [ 1 + ( 2 m − n ) κ ] {\displaystyle N_{\kappa }={\frac {1}{(n-1)!}}\prod _{m=0}^{n}\left[1+(2m-n)\kappa \right]} c n = n κ 2 1 − n 2 κ 2 {\displaystyle c_{n}={\frac {n\kappa ^{2}}{1-n^{2}\kappa ^{2}}}} c n − 1 = 0 {\displaystyle c_{n-1}=0} c n − 2 = n − 1 ( 1 − n 2 κ 2 ) [ 1 − ( n − 2 ) 2 κ 2 ] {\displaystyle c_{n-2}={\frac {n-1}{(1-n^{2}\kappa ^{2})[1-(n-2)^{2}\kappa ^{2}]}}} c m = ( m + 1 ) ( m + 2 ) 1 − m 2 κ 2 c m + 2 for 0 ≤ m ≤ n − 3 {\displaystyle c_{m}={\frac {(m+1)(m+2)}{1-m^{2}\kappa ^{2}}}c_{m+2}\quad {\textrm {for}}\quad 0\leq m\leq n-3}
First member The first member (n = 1 {\displaystyle n=1} κ -Erlang family is the κ -Exponential distributionprobability density function and the cumulative distribution function are defined as:
f κ ( x ) = ( 1 − κ 2 ) exp κ ( − x ) {\displaystyle f_{_{\kappa }}(x)=(1-\kappa ^{2})\exp _{\kappa }(-x)} F κ ( x ) = 1 − ( 1 + κ 2 x 2 + κ 2 x ) exp k ( − x ) {\displaystyle F_{\kappa }(x)=1-{\Big (}{\sqrt {1+\kappa ^{2}x^{2}}}+\kappa ^{2}x{\Big )}\exp _{k}({-x)}}
Second member The second member (n = 2 {\displaystyle n=2} κ -Erlang family has the probability density function and the cumulative distribution function defined as:
f κ ( x ) = ( 1 − 4 κ 2 ) x exp κ ( − x ) {\displaystyle f_{_{\kappa }}(x)=(1-4\kappa ^{2})\,x\,\exp _{\kappa }(-x)} F κ ( x ) = 1 − ( 2 κ 2 x 2 + 1 + x 1 + κ 2 x 2 ) exp k ( − x ) {\displaystyle F_{\kappa }(x)=1-\left(2\kappa ^{2}x^{2}+1+x{\sqrt {1+\kappa ^{2}x^{2}}}\right)\exp _{k}({-x)}}
Third member The second member (n = 3 {\displaystyle n=3} probability density function and the cumulative distribution function defined as:
f κ ( x ) = 1 2 ( 1 − κ 2 ) ( 1 − 9 κ 2 ) x 2 exp κ ( − x ) {\displaystyle f_{_{\kappa }}(x)={\frac {1}{2}}(1-\kappa ^{2})(1-9\kappa ^{2})\,x^{2}\,\exp _{\kappa }(-x)} F κ ( x ) = 1 − { 3 2 κ 2 ( 1 − κ 2 ) x 3 + x + [ 1 + 1 2 ( 1 − κ 2 ) x 2 ] 1 + κ 2 x 2 } exp κ ( − x ) {\displaystyle F_{\kappa }(x)=1-\left\{{\frac {3}{2}}\kappa ^{2}(1-\kappa ^{2})x^{3}+x+\left[1+{\frac {1}{2}}(1-\kappa ^{2})x^{2}\right]{\sqrt {1+\kappa ^{2}x^{2}}}\right\}\exp _{\kappa }(-x)}
The κ -Exponential distribution of type Iκ -Erlang distribution when n = 1 {\displaystyle n=1} A κ -Erlang distribution corresponds to am ordinary exponential distribution when κ = 0 {\displaystyle \kappa =0} n = 1 {\displaystyle n=1}
See also
References
External links
![{\displaystyle f_{_{\kappa }}(x)={\frac {1}{(n-1)!}}\prod _{m=0}^{n}\left[1+(2m-n)\kappa \right]x^{n-1}\exp _{\kappa }(-x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/61cad9e6e1bf0c3690db636ba77df66eb25cc440)
![{\displaystyle F_{\kappa }(x)={\frac {1}{(n-1)!}}\prod _{m=0}^{n}\left[1+(2m-n)\kappa \right]\int _{0}^{x}z^{n-1}\exp _{\kappa }(-z)dz}](https://wikimedia.org/api/rest_v1/media/math/render/svg/75f092d34c654fa496e5afedc7c2fd85351740b9)
![{\displaystyle F_{\kappa }(x)=1-\left[R_{\kappa }(x)+Q_{\kappa }(x){\sqrt {1+\kappa ^{2}x^{2}}}\right]\exp _{\kappa }(-x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/29dcfff1247afb5bd8dc19574e93f2f95cc8ffb6)
![{\displaystyle N_{\kappa }={\frac {1}{(n-1)!}}\prod _{m=0}^{n}\left[1+(2m-n)\kappa \right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/92d7e8a32cc426dfb7e29dfe85bda2d6c1c0e337)
![{\displaystyle c_{n-2}={\frac {n-1}{(1-n^{2}\kappa ^{2})[1-(n-2)^{2}\kappa ^{2}]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4a9b8f0e6e49f42fd4f5c72a25b27d381a52da9c)
![{\displaystyle F_{\kappa }(x)=1-\left\{{\frac {3}{2}}\kappa ^{2}(1-\kappa ^{2})x^{3}+x+\left[1+{\frac {1}{2}}(1-\kappa ^{2})x^{2}\right]{\sqrt {1+\kappa ^{2}x^{2}}}\right\}\exp _{\kappa }(-x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fed65c8cf456f01a8a5ff3c9830b4dcfd470faaa)
