Theory of Statistics - George Mason University

834 Appendix A. Important Probability Distributions
Table A.4. Distributions Useful as Priors for the Normal Parameters
inverted gamma PDF
1
Γ(α)β α
„ 1
y
α, β ∈ IR+ mean 1/β(α − 1) for α > 1
« α+1
e −1/βy I ĪR+ (y)
variance 1/(β 2 (α − 1) 2 (α − 2)) for α > 2
1
inverted chi-squared PDF
Γ(ν/2)2ν/2 „
1
y
ν ∈ IR+ mean 1/(ν − 2) for ν > 2
« ν/2+1
e −1/2y I ĪR+ (y)
variance 2/((ν − 2) 2 (ν − 4)) for ν > 4
inverted Wishart PDF *************************
*************** mean **************
variance ***************
Table A.5. Distributions Derived from the Univariate Normal
lognormal PDF
µ ∈ IR; σ ∈ IR+ mean e µ+σ2 /2
inverse Gaussian PDF
µ, λ ∈ IR+ mean µ
1
√ 2πσ y −1 e −(log(y)−µ)2 /2σ 2
I ĪR+ (y)
variance e 2µ+σ2
(e σ2
− 1)
s
variance µ 3 /λ
skew normal PDF
µ, λ ∈ IR; σ ∈ IR+ mean
q
µ + σ
2λ
λ
2πy 3 e−λ(y−µ)2 /2µ 2 y IĪR+ (y)
1
πσ e−(y−µ)2 /2σ 2
Z λ(y−µ)/σ
e
−∞
−t2 /2
dt
π(1+λ 2 )
variance σ 2 (1 − 2λ 2 /π)
Theory of Statistics c○2000–2013 James E. Gentle