L23-stats-post

L23-stats-post

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Unformatted text preview: informative enough • An 1 − α confidence interval ˆn ˆn is a (random) interval [Θ−, Θ+], ˆn ˆn P(Θ− ≤ θ ≤ Θ+) ≥ 1 − α, s.t. ∀θ – often α = 0.05, or 0.25, or 0.01 – interpretation is subtle Properties: ˆ • E [ Θn ] = θ Confidence intervals (CIs) (consistency) MSE: σ 2/n • CI in estimation of the mean ˆ Θn = (X1 + · · · + Xn)/n – normal tables: Φ(1.96) = 1 − 0.05/2 P • Sample mean often turns out to also be the ML estimate. E.g., if Xi ∼ N (θ, σ 2), i.i.d. ˆ |Θ n − θ | ≤ 1.96 ≈ 0.95 (CLT) √ σ/ n 1.96 σ 1.96 σ ˆ ˆ P Θn − √ ≤ θ ≤ Θn + √ ≈ 0.95 n n More generally: let z be s.t. Φ(z ) = 1 − α/2 zσ zσ ˆ ˆ P Θn − √ ≤ θ ≤ Θn + √...
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This note was uploaded on 06/04/2013 for the course EE 6.431 taught by Professor Johntsitsiklis during the Spring '10 term at MIT.

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