The Kappa Distribution
The Kappa Distribution
This Demonstration shows the probability density function (PDF) and the complementary cumulative distribution function (CCDF) of the distribution, given by
κ
p(x)=
αβ(-β)
α-1
x
exp
κ
α
x
1+
2
β
2
κ
2α
x
and
P
>
exp
κ
α
x
respectively, with and (x) as defined in the Details section. These functions can be viewed as a generalization of the ordinary exponential, which is recovered in the limit as , and present a power-law tail as .
x∈
+
exp
κ
κ→0
x→∞
The exponent quantifies the curvature (shape) of the distribution, which is less pronounced for lower values of the parameter, and more pronounced for higher values. The constant is a characteristic scale, since its value determines the scale of the probability distribution: if is small, then the distribution will be more concentrated around the mode; if is large, then it will be more spread out. Finally, the parameter measures the heaviness of the right tail: the larger its magnitude, the fatter the tail.
α>0
β>0
β
β
κ∈[0,1)
Assuming , the Demonstration shows the effects of different values of the parameters and on the shape of the PDF (on linear scale) and CCDF (both on a linear and log-log scale).
β=1
α
κ
In the last few years the distribution has appeared in a diverse range of applications, including both physical and systems and those in other fields.
κ