7.5.11. Let X₁, X₂,..., Xn, n > 2, be a random sample from the binomial distri- bution b(1,0). 440 (a) Show that Y₁ = X₁ + X₂ + + Xn is a complete sufficient statistic for 0. ... Sufficiency (b) Find the function (Y₁) that is the MVUE of 0. (c) Let Y₂ = (X₁ + X₂)/2 and compute E(Y₂).

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**Statistical Sufficiency and Estimation**

**Problem 7.5.11**

Let \( X_1, X_2, \ldots, X_n \), where \( n > 2 \), be a random sample from the binomial distribution \( b(1, \theta) \).

**Tasks:**

(a) Show that \( Y_1 = X_1 + X_2 + \cdots + X_n \) is a complete sufficient statistic for \( \theta \).

(b) Find the function \( \varphi(Y_1) \) that is the MVUE (Minimum Variance Unbiased Estimator) of \( \theta \).

(c) Let \( Y_2 = (X_1 + X_2)/2 \) and compute \( E(Y_2) \).

(d) Determine \( E(Y_2 | Y_1 = y_1) \).
Transcribed Image Text:**Statistical Sufficiency and Estimation** **Problem 7.5.11** Let \( X_1, X_2, \ldots, X_n \), where \( n > 2 \), be a random sample from the binomial distribution \( b(1, \theta) \). **Tasks:** (a) Show that \( Y_1 = X_1 + X_2 + \cdots + X_n \) is a complete sufficient statistic for \( \theta \). (b) Find the function \( \varphi(Y_1) \) that is the MVUE (Minimum Variance Unbiased Estimator) of \( \theta \). (c) Let \( Y_2 = (X_1 + X_2)/2 \) and compute \( E(Y_2) \). (d) Determine \( E(Y_2 | Y_1 = y_1) \).
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