Let 20 21 22 Be A Sequence Of Independent Random Variables With Expectation 0 And Variance 1 For N 1 Define The 1 (63.08 KiB) Viewed 31 times
Let 20, 21, 22, ... be a sequence of independent random variables with expectation 0 and variance 1. For n> 1 define the random variables Xn = Zn + {Zn-1. = ) (a) Determine cov(Xn, Xn+k) for all n>1 and k > 0. [3 marks] (b) Let U and V be two random variables with mean 0 and non-zero variance. Define the function f :R → [0,00) such that f(c) = E((U – CV)?). Show that f is minimised at c= 0 if and only if U and V are uncorrelated. [2 marks] = =
(c) We aim to determine (a*, b*) € R2 that minimises E [(X3 – aXı – 6X2)?]. Using (b) show that if (a*,b*) minimises E [(X3 – aX1 – 6X2)?], then = E [X;(X3 – a*X1 – b* X2)] = 0, j = 1, 2. (1 mark] (d) Assuming the existence of a minimiser, determine (a*,b*) which minimises E[(X3 – aX1 – 6X2)?). (1 mark]
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