1. Let ,...,n be realizations of independent random variables X,..., Xn with PDF fx(x; 0) =...

 

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1. Let ₁,...,n be realizations of independent random variables X₁,..., Xn with PDF fx(x; 0) = 30³ x ¹1(x > 0), where > 0 is a statistical parameter and 3> 0 is known. (a) (5 marks) Show that if X~ fx (0) then E[X] = 0 and E[X²] = 30². (b) (5 marks) Derive the method of moments estimator MM of 0, and deduce its variance. (c) (2 marks) Show that the maximum likelihood estimate of is min{₁,..., In}. (d) (3 marks) Show that the CDF of X~ fx(0) is 3 (9) ³ Fx(x) = (f) (5 marks) Show that - (e) (5 marks) Show that the CDF of the maximum likelihood estimator with n observa- tions is F(m; 0) = x ≥ 0, x < 0. 3n (2) ³ m≥ 0. m < 0. 3n 37 (0ML - 0)→D(0) Y~1 Exponential(1), where OML is the maximum likelihood estimator with n observations. 1. Let ₁,...,n be realizations of independent random variables X₁,..., Xn with PDF fx(x; 0) = 30³ x ¹1(x > 0), where > 0 is a statistical parameter and 3> 0 is known. (a) (5 marks) Show that if X~ fx (0) then E[X] = 0 and E[X²] = 30². (b) (5 marks) Derive the method of moments estimator MM of 0, and deduce its variance. (c) (2 marks) Show that the maximum likelihood estimate of is min{₁,..., In}. (d) (3 marks) Show that the CDF of X~ fx(0) is 3 (9) ³ Fx(x) = (f) (5 marks) Show that - (e) (5 marks) Show that the CDF of the maximum likelihood estimator with n observa- tions is F(m; 0) = x ≥ 0, x < 0. 3n (2) ³ m≥ 0. m < 0. 3n 37 (0ML - 0)→D(0) Y~1 Exponential(1), where OML is the maximum likelihood estimator with n observations.

Answer rating: 100% (QA)

To solve this problem well go through each part step by step a Finding the expected value EX and EX The expected value EX is given by EX xfXxdx Substituting the given PDF fXx into the integral EX x 3x... View the full answer