BUILT-IN MATHEMATICA SYMBOL
StableDistribution
- A linear combination of independent identically distributed stable random variables is also stable.
- A stable distribution is defined in terms of its characteristic function
, which satisfies a functional equation where for any 
and 
there exist 
and 
such that 
. The general solution to the functional equation has four parameters.
- StableDistribution allows 0<
≤2, 
, 
to be any real number, and 
to be any positive real number.
- CharacteristicFunction[StableDistribution[0, , ...], t] is continuous in and given by
![exp(i mu t-sigma TemplateBox[{t}, Abs] (1+2i beta/pi sgn(t) log(TemplateBox[{{t, , sigma}}, Abs]))) alpha=1; exp(i mu t-TemplateBox[{{t, , sigma}}, Abs]^alpha (1+i beta tan((pi alpha)/2) sgn(t) (TemplateBox[{{t, , sigma}}, Abs]^(1-alpha)-1))) alpha!=1](Files/StableDistribution.en/9.png "exp(i mu t-sigma TemplateBox[{t}, Abs] (1+2i beta/pi sgn(t) log(TemplateBox[{{t, , sigma}}, Abs]))) alpha=1; exp(i mu t-TemplateBox[{{t, , sigma}}, Abs]^alpha (1+i beta tan((pi alpha)/2) sgn(t) (TemplateBox[{{t, , sigma}}, Abs]^(1-alpha)-1))) alpha!=1")
.
- CharacteristicFunction[StableDistribution[1, , ...], t] is discontinuous in and given by
![exp(i mu t-sigma TemplateBox[{t}, Abs] (1+2 i beta/pi sgn(t) log(TemplateBox[{t}, Abs]))) alpha=1; exp(i mu t-TemplateBox[{{t, , sigma}}, Abs]^alpha (1-i beta tan((pi alpha)/2) sgn(t))) alpha!=1](Files/StableDistribution.en/10.png "exp(i mu t-sigma TemplateBox[{t}, Abs] (1+2 i beta/pi sgn(t) log(TemplateBox[{t}, Abs]))) alpha=1; exp(i mu t-TemplateBox[{{t, , sigma}}, Abs]^alpha (1-i beta tan((pi alpha)/2) sgn(t))) alpha!=1")
.
- StableDistribution[] is equivalent to StableDistribution[1, , 0, 0, 1].
- StableDistribution[,
] is equivalent to StableDistribution[1, , , 0, 1].
- StableDistribution[, , , ] is equivalent to StableDistribution[1, , , , ].
- StableDistribution can be used with such functions as Mean, CDF, and RandomVariate.
New in 8
with index of stability 
:
