Math Problem Statement
Låt X1,X2,...,Xn vara oberoende och likafördelade med Exp(β)-fördelningen, dvs med täthetsfunktionen
f(x)=1/βe^(-x/β) för 0<=x<∞.
om man ska konstruera ett konfidensintervall för β. Vad skulle vare en lämplig referensvariabel med tillhörande fördelning?
Ett av dessa svarsalternativ stämmer:
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( X(streck)_n - 𝛃 ) ≈ N ( 0, 1 )
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( X(streck)_n - 𝛃 ) ~ Exp ( 1 )
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sqrt(n) * ( X(streck)_n - 𝛃 ) ≈ N ( 0, 1 )
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( ( X(streck)_n - 𝛃 ) / ( ( X(streck)_n ) / sqrt(n) ) ) ~ Exp (1)
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sqrt(n) * ( X(streck)_n - 𝛃 ) ~ Exp (1)
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( ( X(streck)_n - 𝛃 ) / ( ( X(streck)_n ) / sqrt(n) ) ) ≈ N ( 0, 1 )
Solution
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Math Problem Analysis
Mathematical Concepts
Probability Distributions
Exponential Distribution
Central Limit Theorem
Confidence Intervals
Formulas
Exponential density function: f(x) = 1/β e^(-x/β), for x ≥ 0
Sample mean: X̄_n = (1/n) ∑ X_i
Standardized variable: sqrt(n) * (X̄_n - β) ≈ N(0, 1)
Theorems
Central Limit Theorem
Properties of Exponential Distribution
Suitable Grade Level
Undergraduate Level
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