(a) Take a parameter 0 € R and a function 0 : R + R. Take a square-integrable random variable X(0), depending on 0, that

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A Take A Parameter 0 R And A Function 0 R R Take A Square Integrable Random Variable X 0 Depending On 0 That 1 (26.69 KiB) Viewed 23 times

(a) Take a parameter 0 € R and a function 0 : R + R. Take a square-integrable random variable X(0), depending on 0, that has g(2;4) > 0 as its density function and assume g to be differentiable in 6. Define L(x; ) as the Log-Likelihood function associated to g in 0. (i) Show that E[L(X(0);0)] = 0. [4 marks) (ii) Given be R, define Z := 4(X(0))L(X(0);0) – bL(X(0);0). Determine the b that minimizes Var(Z”), the variance of the control variate Z”. [6 marks]