3.3.4 Beta Distribution
The beta(α, β) pdf is
f (x|α, β) = 1
B(α, β)xα−1(1 − x)β−1, 0 < x < 1, α > 0, β > 0, where B(α, β) denotes the beta function,
B(α, β) = Z 1
0
xα−1(1 − x)β−1dx = Γ(α)Γ(β) Γ(α + β). For n > −α, we have
EXn= 1 B(α, β)
Z 1
0
xnxα−1(1 − x)β−1dx
= B(α + n, β)
B(α, β) = Γ(α + n)Γ(α + β) Γ(α + β + n)Γ(α). Then mean and variance are
EX = α
α + β and VarX = αβ
(α + β)2(α + β + 1).
Genesis: Handout 1.
3.3.5 Cauchy Distribution
The Cauchy distribution is a symmetric, bell-shaped distribution on (−∞, ∞) with pdf
f (x|θ) = 1 π
1
1 + (x − θ)2, −∞ < x < ∞, −∞ < θ < ∞.
The mean of Cauchy distribution does not exist, that is,
E|X| = Z ∞
−∞
1 π
|x|
1 + (x − θ)2dx = ∞.
Since E|X| = ∞, it follows that no moments of the Cauchy distribution exist. In particular, the mgf does not exist.
Genesis: Handout 2:
1
3.3.6 Lognormal Distribution
If X is a random variable whose logarithm is normally distributed, then X has a lognormal distribution. The pdf of X can be obtained by straightforward transformation of the normal pdf, yielding
f (x|µ, σ2) = 1
√2πσ 1
xe−(log x−µ)2/(2σ2), 0 < x < ∞, −∞ < µ < ∞, σ > 0, for the lognormal pdf.
EX = Eelog X = EeY = eµ+(σ2/2).
VarX = e2(µ+σ2)− e2µ+σ2.
Genesis: Handout 3.
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