Mathematics · Probability and Statistics

Let X 1 , X 2 , ..., X n be independent and identically distributed (i.i.d.) N(θ, 1) random variables. Suppose the prior distribution of θ is N(0, σ 2 ). Under a squared error loss function, let X = ∑ i=1 n X i n . Then, the Bayes estimator of θ is:

Verified Solution4 OptionsUpdated: October 2026👁️1 View

1X 1 + σ 2
2X
3n X σ 2 1 + n σ 2✓
4n( X + σ 2 ) 1 + n σ 2

Correct answer

n X σ 2 1 + n σ 2

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Let X 1 , X 2 , ..., X n be independent and identically distributed (i.i.d.) N(θ, 1) random variables. Suppose the prior distribution of θ is N(0, σ 2 ). Under a squared error loss function, let X = ∑ i=1 n X i n . Then, the Bayes estimator of θ is:

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Let X 1 , X 2 , ..., X n be independent and identically distributed (i.i.d.) N(&theta;, 1) random variables. Suppose the prior distribution of &theta; is N(0, &sigma; 2 ). Under a squared error loss function, let X = &sum; i=1 n X i n . Then, the Bayes estimator of &theta; is:

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Let X 1 , X 2 , ..., X n be independent and identically distributed (i.i.d.) N(θ, 1) random variables. Suppose the prior distribution of θ is N(0, σ 2 ). Under a squared error loss function, let X = ∑ i=1 n X i n . Then, the Bayes estimator of θ is: