PDF of Lecture Notes - School of Mathematical Sciences
PDF of Lecture Notes - School of Mathematical Sciences
PDF of Lecture Notes - School of Mathematical Sciences
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1. DISTRIBUTION THEORY<br />
2. If Z 1 , Z 2 , . . . , Z r are i.i.d. N(0, 1) and Z = (Z 1 , . . . , Z r ) T ,<br />
⇒f Z (z) =<br />
r∏<br />
i=1<br />
1<br />
√<br />
2π<br />
e −z2 i /2<br />
=<br />
which is N r (0 r , I r×r ) <strong>PDF</strong>.<br />
Theorem. 1.10.1<br />
1<br />
(2π) r/2 e−1/2 P r<br />
i=1 z2 i =<br />
1<br />
(2π) r/2 e− 1 2 zT z ,<br />
Suppose X ∼ N r (µ, Σ) and let Y = AX + b, where b ∈ R r and A r×r invertible are<br />
fixed. Then Y ∼ N r (Aµ + b, AΣA T ).<br />
Pro<strong>of</strong>.<br />
We use the transformation rule for <strong>PDF</strong>s:<br />
If X has joint <strong>PDF</strong> f X (x) and Y = g(X) for g : R r → R r invertible, continuously<br />
differentiable, ( ) then f Y (y) = f X (h(y))|H|, where h : R r → R r such that h = g −1 and<br />
∂hi<br />
H = .<br />
∂y j<br />
In this case, we have g(x) = Ax + b<br />
⇒h(y) = A −1 (y − b)<br />
[ solving for x in<br />
y = Ax + b<br />
]<br />
⇒H = A −1 .<br />
Hence,<br />
f Y (y) =<br />
=<br />
=<br />
1<br />
(2π) r/2 |Σ| 1/2 e−1/2(A−1 (y−b)−µ) T Σ −1 (A −1 (y−b)−µ) |A −1 |<br />
1<br />
(2π) r/2 |Σ| 1/2 |AA T | 1/2 e−1/2(y−(Aµ+b))T (A −1 ) T Σ −1 (A −1 )(y−(Aµ+b))<br />
1<br />
(2π) r/2 |AΣA T | 1/2 e−1/2(y−(Aµ+b))T (AΣA T ) −1 (y−(Aµ+b))<br />
which is exactly the <strong>PDF</strong> for N r (Aµ + b, AΣA T ).<br />
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