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(a) Find a sufficient statistic T(X₁,..., Xn) for p. (b) Show that 1 {X₁ = 1, X₂ = 0} is an unbiased estimator of p(1-p), where 1 {} denotes the indicator function. (c) Use the Rao-Blackwell theorem to improve the above estimator.

(a) Find a sufficient statistic T(X₁,..., Xn) for p. (b) Show that 1 {X₁ = 1, X₂ = 0} is an unbiased estimator of p(1-p), where 1 {} denotes the indicator function. (c) Use the Rao-Blackwell theorem to improve the above estimator.

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter1: Functions
Section1.2: The Least Square Line
Problem 1E
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Suppose X₁,..., Xn when n > 2 are i.i.d Bernoulli(p) where 0 < p < 1 is unknown. (a) Find a sufficient statistic T(X₁,..., Xn) for p. (b) Show that 1 {X₁ = 1, X₂ = 0} is an unbiased estimator of p(1-p), where 1 {} denotes the indicator function. (c) Use the Rao-Blackwell theorem to improve the above estimator.
Transcribed Image Text:Suppose X₁,..., Xn when n > 2 are i.i.d Bernoulli(p) where 0 < p < 1 is unknown. (a) Find a sufficient statistic T(X₁,..., Xn) for p. (b) Show that 1 {X₁ = 1, X₂ = 0} is an unbiased estimator of p(1-p), where 1 {} denotes the indicator function. (c) Use the Rao-Blackwell theorem to improve the above estimator.
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