PDF of Lecture Notes - School of Mathematical Sciences

1. DISTRIBUTION THEORY
⇒ det(G) = z(1 − y) − (−zy)
where z > 0. Why
= z(1 − y) + zy = z,
f Y,Z (y, z) = f X1 (yz)f X2
(
(1 − y)z
)
z
Γ(α) (yz)α−1 e −λyz λβ ( ) β−1e
(1 − y)z −λ(1−y)z z
Γ(β)
= λα
=
1
Γ(α)Γ(β) λα+β y α−1 (1 − y) β−1 z α+β−1 e −λz .
Step 3:
f Y (y) =
∫ ∞
0
f(y, z) dz
=
=
∫
Γ(α + β)
∞
Γ(α)Γ(β) yα−1 (1 − y) β−1
Γ(α + β)
Γ(α)Γ(β) yα−1 (1 − y) β−1
0
λ α+β z α+β−1
Γ(α + β) e−λz dz
= Beta(α, β), for 0 < y < 1.
Exercise: Justify the range of values for y.
1.9 Moments
Suppose X 1 , X 2 , . . . , X r are RVs. If Y = h(X) is defined by a real-valued function h,
then to find E(Y ) we can:
1. Find the distribution of Y .
⎧∫ ∞
yf Y (y) dy
⎪⎨ −∞
2. Calculate E(Y ) =
∑
⎪⎩ yp(y)
y
continuous
discrete
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