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# zz-26 - Utah State University ECE 6010 Stochastic Processes...

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Utah State University ECE 6010 Stochastic Processes Homework # 4 Solutions 1. Suppose X ∼ N ( μ , Σ). (a) Show that E [ X ] = μ and cov( X , X ) = Σ. Using characteristic functions: φ X ( u ) = exp[ i u T μ - 1 2 u T Σ u ] Taking the gradient with respect to u we have ∂φ X ( u ) u = exp[ i u T μ - 1 2 μ T Σ μ ]( i μ - Σ u ) Now evaluating at u = 0 and dividing by i we obtain E [ X ] = μ . Taking the gradient again with respect to u we have exp[ i u T μ - 1 2 μ T Σ μ ]( - Σ) + exp[ i u T μ - 1 2 μ T Σ μ ]( i μ - Σ u )( i μ - Σ u ) T and evaluating at u = 0 we have - Σ - μμ Dividing by i - 2 we obtain E [ X 2 ] = Σ+ μμ T , from which it follows that cov( X , X ) = Σ. (b) Show that A X + b ∼ N ( A μ + b , A Σ A T ). We note that φ X ( u ) = E [ e i u T X ] = exp[ i u T μ - 1 2 μ T Σ μ ]. Then φ Y ( u ) = E [ e i u T Y ] = E [ e i u T ( A X + b ) ] = e i u T b E [ e i u T A X ] But we can compute the expected value using what we know from φ X ( u ): E [ e i u T A X ] = φ X ( v ) | v = A T u = exp[ i u T A μ - 1 2 u T A Σ A T u ] 1

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So φ Y ( u ) = e i u T b exp[ i u T A μ - 1 2 u T A Σ A T u ] = exp[ i u ( A μ + b ) - 1 2 u T ( A Σ A T ) u ] . From the form of the characteristic function we identify that Y is Gaussian with E [ Y ] = A μ + b cov( Y , Y ) = A Σ A T . (c) Suppose Σ > 0 and write Σ = CC T . Show that C - 1 ( X - μ ) ∼ N ( 0 , I ). If Y = C - 1 X - C - 1 μ we have, by applying the previous result E [ Y ] = C - 1 μ - C - 1 μ = 0 cov( Y , Y ) = C - 1 Σ C - T = C - 1 ( CC T ) C - T = I.
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zz-26 - Utah State University ECE 6010 Stochastic Processes...

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