Jointly Gaussian random variables, MMSE and linear MMSE estimation
نویسنده
چکیده
• Proof: X is j G implies that V = uX is G with mean uμ and variance uΣu. Thus its characteristic function, CV (t) = e ituμe−t 2uTΣu/2. But CV (t) = E[e itV ] = E[e TX ]. If we set t = 1, then this is E[e TX ] which is equal to CX(u). Thus, CX(u) = CV (1) = e iuμe−u TΣu/2. • Proof (other side): we are given that the charac function ofX, CX(u) = E[eiuTX ] = e μe−u TΣu/2. Consider V = uX. Thus, CV (t) = E[e itV ] = CX(tu) = e iuμe−t 2uTΣu/2. Also, E[V ] = uμ, var(V ) = uΣu. Thus V is G.
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