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 />
⎧<br />
⎪⎨<br />
⎪⎩<br />
How did this come about:<br />
[ ( ) a b x<br />
(x, y)<br />
c d]<br />
y<br />
= (xy)<br />
[ ] ax + by<br />
cx + dy<br />
= x(ax + by) + y(cx + dy)<br />
= ax 2 + bxy + cyx + dy 2 ⎫⎪ ⎬<br />
⎪ ⎭<br />
= ( (x 1 − µ 1 ) − Σ 12 Σ −1<br />
22 (x 2 − µ 2 ) ) T<br />
V<br />
−1 ( (x 1 − µ 1 ) − Σ 12 Σ −1<br />
22 (x 2 − µ 2 ) )<br />
+ (x 2 − µ 2 ) T Σ −1<br />
22 (x 2 − µ 2 )<br />
⎧<br />
⎪⎨<br />
Note:<br />
(x − y) T A(x − y) = (x − y) T (Ax − Ay)<br />
= x T Ax − x T Ay − y T Ax + y T Ay<br />
⎪⎩<br />
And: Σ T 12 = Σ 21<br />
⎫⎪ ⎬<br />
⎪ ⎭<br />
= ( x 1 − (µ 1 + Σ 12 Σ −1<br />
22 (x 2 − µ 2 )) ) T (<br />
Σ11 − Σ 12 Σ −1<br />
22 Σ 21<br />
) −1×<br />
(<br />
x1 − (µ 1 + Σ 12 Σ −1<br />
22 (x 2 − µ 2 )) ) + (x 2 − µ 2 ) T Σ −1<br />
22 (x 2 − µ 2 ).<br />
Combining (1), (2) we obtain:<br />
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