I Let X, X2,. , Xn be a random sample from a population with probability mass...

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I Let X₁, X2,. , Xn be a random sample from a population with probability mass func- tion given by f(x) = S0/2 1-0 0 if x = 0,1 if x = 2 otherwise 2 where € (0, 1) is an unknown parameter. Define N₂(X) as the number of X₂'s that result in the value 2. Also define the statistics T₁(X) (4 - 2X)/3 and T₂(X) 1 - N₂(X)/n. = = (i) Show that E(T;(X)) = 0, j = 1, 2, that is, T₁(X) and T₂(X) are unbiased estima- tors of 0. (ii) Show that N₂(X) is a minimal sufficient statistic for 0. (iii) Is N₂(X) a complete statistic? Explain your answer. (iv) Show that X is not sufficient for 0. (v) Compute and compare the variances Var(T₁ (X)) and Var(T₂(X)). I Let X₁, X2,. , Xn be a random sample from a population with probability mass func- tion given by f(x) = S0/2 1-0 0 if x = 0,1 if x = 2 otherwise 2 where € (0, 1) is an unknown parameter. Define N₂(X) as the number of X₂'s that result in the value 2. Also define the statistics T₁(X) (4 - 2X)/3 and T₂(X) 1 - N₂(X)/n. = = (i) Show that E(T;(X)) = 0, j = 1, 2, that is, T₁(X) and T₂(X) are unbiased estima- tors of 0. (ii) Show that N₂(X) is a minimal sufficient statistic for 0. (iii) Is N₂(X) a complete statistic? Explain your answer. (iv) Show that X is not sufficient for 0. (v) Compute and compare the variances Var(T₁ (X)) and Var(T₂(X)).