PDF of Lecture Notes - School of Mathematical Sciences

1. DISTRIBUTION THEORY
Recall standard limit:
(
lim 1 + a ) n
= e
a
n→∞ n
⇒ f T (t) → 1 √
2π
e −t2
2 as k → ∞.
Suppose h : R r → R r is continuously differentiable.
That is,
h(x 1 , x 2 , . . . , x r ) = (h 1 (x 1 , . . . , x r ), h 2 (x 1 , . . . , x r ), . . . , h r (x 1 , . . . , x r )) ,
where each h i (x) is continuously differentiable. Let
⎡
⎤
∂h 1 ∂h 1 ∂h 1
. . .
∂x 1 ∂x 2 ∂x r
∂h 2 ∂h 2 ∂h 2
. . .
∂x
H = 1 ∂x 2 ∂x r
. . . . . .
⎢
⎥
⎣∂h r ∂h r ∂h r
⎦
. . .
∂x 1 ∂x 2 ∂x r
If H is invertible for all x, then ∃ an inverse mapping:
g : R r → R r
g(h(x)) = x.
with property that:
It can be proved that the matrix of partial derivatives
( ) ∂gi
G = satisfies G = H −1 .
∂y j
Theorem. 1.8.4
Suppose X 1 , X 2 , . . . , X r have joint PDF f X (X 1 , . . . , X r ), and let h, g, G be as above.
If Y = h(X), then Y has joint PDF
Remark:
f Y (y 1 , y 2 , . . . , y r ) = f X
(
g(y)
)
| det G(y)|
Can sometimes use H −1 instead of G, but need to be careful to evaluate H −1 (x) at
x = h −1 (y) = g(y).
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