Quiz4 - 36-226 Summer 2010 Quiz 4 July 30 1. Indicate...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 36-226 Summer 2010 Quiz 4 July 30 1. Indicate whether the following are TRUE or FALSE. Write the entire word below each statement. Do not assume ANYTHING not written in the problem statement. (a) If I fail to reject the null hypothesis with α = 0.01 then I can reject the null with α = 0.05. FALSE (b) The p-value is the probability that the alternative hypothesis is true given the data I have seen. FALSE (c) If {X > 2} is the rejection region for an MP test of the hypothesis H0 : θ = 0 versus Ha : θ = 3, then this is also the rejection region for a UMP test of the hypothesis H0 : θ = 0 versus Ha : θ > 0. TRUE (d) Let X ∼ N(µ, 1). There is no UMP test for the hypothesis H0 : µ = 0 versus Ha : µ = 0. TRUE (e) If I fail to reject the null hypothesis, then the p-value was greater than α. TRUE 2. Let X1 , . . . , Xn be i.i.d. U(0, θ). Show that the Pareto distribution with density p(θ) = aba I (θ) θa+1 [b,∞) is conjugate and find the new parameters a and b of the posterior distribution. Hint: Remember to keep around EVERYTHING that depends on θ when you proportion things away. Hint 2: Think about indicator functions as logicals and remember that in the posterior, X is fixed and θ is random. 1 I (X ) θn [0,θ] (n) 1 p(θ) ∝ a+1 I[b,∞) (θ) θ 1 1 p(θ | X1 , . . . , Xn ) ∝ n I[0,θ] (X(n) ) a+1 I[b,∞) (θ) θ θ 1 ∝ n+a+1 I[max{b,X(n) },∞) (θ) θ ∝ Pareto(a = a + n, b = max{b, X(n) }) L(θ) = So the Pareto is conjugate and the posterior has new parameters a = a + n and b = max{b, X(n) }. 1 ...
View Full Document

This document was uploaded on 07/14/2011.

Ask a homework question - tutors are online