Quiz3 - 36-226 Summer 2010 Quiz 3 July 23 1(2 points What...

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Unformatted text preview: 36-226 Summer 2010 Quiz 3 July 23 1. (2 points) What is a pivotal quantity? A pivotal quantity satisﬁes the following two points: • It is a function of the data and the unknown parameter(s) • It has a distribution which does not depend on any unknown parameters 2. (1 point each) Let X ∼ exp(θ) and let Y1 , . . . , Yn be iid exp(θ). Which of the following are pivotal quantities? You may use the following facts: 1 • fX (x) = e−x/θ θ • FX (x) = 1 − e−x/θ 2 X ∼ χ2 (2) θ 2n • Y ∼ χ2 (2n) θ • (a) 1 − e−X/θ Yes (b) 2Y No Y −θ √ if n is large. (c) s/ n Yes 3. (5 points) Let X ∼ exp(θ). Consider the hypothesis test H0 : θ = 1/5 versus Ha : θ = 1/2. A size α = 0.05 test for this hypothesis is X in the rejection region R = [− 1 ln 0.05, ∞), i.e. 5 1 reject the null if we observe X greater than c = − 5 ln 0.05. Calculate the probabilities of Type I and Type II error. Notice that the PDF and CDF of X are given in problem 2. P(Type I error) = PH0 (X ∈ R) = α = 0.05. P(Type II error) = PHa (X ∈ R) = FX (c | θ = 1/2) = 1 − e−2c =1−e 2 5 ln 0.05 = 1 − (0.05)2/5 ≈ 0.70 1 (you can stop here) ...
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This document was uploaded on 07/14/2011.

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