Let X 1 , X 2 , , X n denote the outcomes of a series of n independent trials, where for i 1, 2, , n Let X X 1 X 2 X n (a) Show that p 1 X 1 and p 2 X n are unbiased estimators for p (b) Intuitively, p 2 is a better estimator than p 1 because p 1 fails to include any of the information about the parameter contained in trials 2 through n Verify that speculation by comparing the variances of p 1 and p 2 1 with probabilityp 0 with probability 1 p

Question: Let X 1 , X 2 , . . . , X n denote the outcomes of a series of n independent trials, where for

Let X₁, X₂, . . . , X_n denote the outcomes of a series of n independent trials, where

1 with probabilityp 0 with probability 1 p

for i =1, 2, . . . , n. Let X = X₁ + X₂ +・・・+ X_n.

(a) Show that p̂₁ = X₁ and p̂₂ = X/n are unbiased estimators for p.

(b) Intuitively, p̂₂ is a better estimator than p̂₁ because p̂₁ fails to include any of the information about the parameter contained in trials 2 through n. Verify that speculation by comparing the variances of p̂₁ and p̂₂.

1 with probabilityp 0 with probability 1 p

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