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## Monday, February 2, 2015

### 2 multivariat normal variables can be indie

Lawler's examples show that given iid standard normals Z1, Z2 ..., two "composed" random vars "X" and "Y" can be made independent of each other by adjusting their composition multipliers a b c d:

X:= a Z1 + b Z2
Y:= c Z1 + d Z2

(Simplest example -- X:= Z1 + Z2 and Y:= Z1 - Z2. See Lawler's notes P39.
Note X = Y + 2*Z2 so they look like related but actually independent!)

This independence is counter intuitive. I'm stilling look out for an intuitive interpretation.

Note X is never independent of Z1.

For 2 joint normal RVs (and only joint normals), 0 correlation implies independence.... Therefore, we only need to show E[XY] = E[X]E[Y]. In our simple example, RHS = 0*0 and

LHS: E[XY] := E[ (Z1+Z2)(Z1-Z2) ] = E[ Z1 Z1 ] - E[ Z2 Z2 ] = 0, since the 2 terms have identical expectations.

A classic counter-example. There's a textbook on counter-examples in calculus, in which the authors argued for the importance of counter examples.