PDF of Lecture Notes - School of Mathematical Sciences

1. DISTRIBUTION THEORY
Figure 12: Area A
3. Dirichlet distribution is defined by PDF
f(x 1 , x 2 , . . . , x r ) = Γ(α 1 + α 2 + · · · + α r + α r+1 )
Γ(α 1 )Γ(α 2 ) . . . Γ(α r )Γ(α r+1 ) ×
x α 1−1
1 x α 2−1
2 . . . x αr−1
r (1 − x 1 − · · · − x r ) α r+1
for x 1 , x 2 , . . . , x r > 0,
∑
xi < 1, and parameters α 1 , α 2 , . . . , α r+1 > 0.
#
Recall joint PDF:
∫
P (X ∈ A) =
Note: the joint PDF must satisfy:
∫
. . . f(x 1 , . . . , x r ) dx 1 . . . dx r .
A
1. f(x) ≥ 0 for all x;
2.
∫ ∞
. . .
∫ ∞
−∞ −∞
f(x 1 , . . . , x r ) dx 1 . . . dx r = 1.
Definition. 1.7.6
( )
X1
If X = has joint PDF f(x) = f(x
X 1 , x 2 ), then the marginal PDF of X 1 is given
2
by:
∫ ∞ ∫ ∞
f X1 (x 1 ) = . . . f(x 1 , . . . , x r1 , x r1 +1, . . . x r ) dx r1 +1dx r1 +2 . . . dx r ,
−∞ −∞
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