STA414H1 Quiz: hw1-sol
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Cov(x, y ) = e(x e(x))(y e(y ))t. = e(xy t ) e(x e(y t )) e(e(x)y t ) + e(e(x)e(y t )) = e(x)e(y t ) e(x)e(y t ) e(x)e(y t ) + e(x)e(y t ) E(x + ay ) = e([xi] + [(ay )i]) V ar(x + ay ) = v ar([xi] + [ayi]) = v ar(x) + e(ay e(ay ))(ay e(ay ))t. = v ar(x) + e(a(y e(y )))(a(y e(y )))t. = v ar(x) + ae(y e(y ))(y e(y ))t at. = v ar(x) + av ar(y )at c) Since any linear combination of gaussians are again a gaussian, the resulting random vector is also a. Its distribution is completely determined by its mean and variance. For example, the probability density function of a random value uniformly distributed between 0. 2 has value 2 in the interval [0, 1. 2 ] such that the volume over the domain integrates to one. b) p(x) = (x )2.
