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a) It is known that the random variable Y has a probability density function f; 0) = {o, SØy®-1, 0

a) It is known that the random variable Y has a probability density function f; 0) = {o, SØy®-1, 0

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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QUESTION 4 a) It is known that the random variable Y has a probability density function (Øyº-1, 0<y< 1 lo, f(y;0) = elsewhere It is desired to test the null hypothesis Ho: 0 = 2 against the alternative H,:0 = 1. A random sample Y, of size n = 1 is to be used. i) Calculate the level of significance or type I error which rejects Ho if Y, <-. 3|Page ii) Consider the test which rejects Ho if Y, < c. If the level of significance for the test is 0.1. What is the value of c? iii) Show that the product [I Yi is sufficient statistic for 0 using the factorisation method. iv) Show that the product II1 Yi is minimal sufficient for 0. b) Let Y1, Y2, ..., Yn be a random sample from a population with probability density function in part a). Show that the best test for the hypothesis in part a) rejects Ho if |yi <c i=1 where c solves the probability equation a = P(II1Yis c[0 = 2). c) Let X1,X2, ..., Xn be a random sample from GAMMA(2,ß) distribution, and consider Y = E-,Xi- Show whether or not Y is a pivotal quantity and give its distribution.
Transcribed Image Text:QUESTION 4 a) It is known that the random variable Y has a probability density function (Øyº-1, 0<y< 1 lo, f(y;0) = elsewhere It is desired to test the null hypothesis Ho: 0 = 2 against the alternative H,:0 = 1. A random sample Y, of size n = 1 is to be used. i) Calculate the level of significance or type I error which rejects Ho if Y, <-. 3|Page ii) Consider the test which rejects Ho if Y, < c. If the level of significance for the test is 0.1. What is the value of c? iii) Show that the product [I Yi is sufficient statistic for 0 using the factorisation method. iv) Show that the product II1 Yi is minimal sufficient for 0. b) Let Y1, Y2, ..., Yn be a random sample from a population with probability density function in part a). Show that the best test for the hypothesis in part a) rejects Ho if |yi <c i=1 where c solves the probability equation a = P(II1Yis c[0 = 2). c) Let X1,X2, ..., Xn be a random sample from GAMMA(2,ß) distribution, and consider Y = E-,Xi- Show whether or not Y is a pivotal quantity and give its distribution.
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