Student's t-distribution
We start with a zero skew probability density (a more complicated version for skew-mixture will be shown in a subsequent post) of:
f(x) = Γ[(γ+1)/2] / [β*√(πγ)*Γ(γ/2)] * [1+1/γ*((x-α)/β)^2]^[-(γ+1)/2]
Gamma, Γ(z), is the positive integral of t^(z-1)*e^(-t)dt.
Average at γ<1 is undefined. The shape parameter is γ.
Variance at γ=1 is undefined and follows the Cauchy distribution
Variance at γ=2 is infinity
We start with a zero skew probability density (a more complicated version for skew-mixture will be shown in a subsequent post) of:
f(x) = Γ[(γ+1)/2] / [β*√(πγ)*Γ(γ/2)] * [1+1/γ*((x-α)/β)^2]^[-(γ+1)/2]
Gamma, Γ(z), is the positive integral of t^(z-1)*e^(-t)dt.
Average at γ<1 is undefined. The shape parameter is γ.
Average at γ>1 is location parameter α
Variance at γ=2 is infinity
Variance at γ>2 is β^2*γ/(γ-2). The scale parameter is β.
Skew at γ<3 is undefined
Skew at γ>3 is 0
Excess kurtosis (leptokurtosis) at γ<4 is undefined
Excess kurtosis (leptokurtosis) at γ>4 is 6/(γ-4)
Excess kurtosis at γ→infinity limits to 0
Tail distribution at finite γ follows the "power law" at power 1/(γ+1). For example, how much greater is the event probability of a Cauchy versus at γ=3?
It is the square root of the probability, as a power of 1/(1+1)=1/2 is the root of a power of 1/(3+1)=1/4.
Check back on these pages regularly as we expand different formulae
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