class8 q

# class8 q - THE UNIVERSITY OF HONG KONG DEPARTMENT OF...

This preview shows pages 1–2. Sign up to view the full content.

THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT 1302 PROBABILITY AND STATISTICS II EXAMPLE CLASS 8 1. Let X 1 and X 2 be i.i.d. exponential random variables with mean θ . It is well known that S = X 1 + X 2 has a gamma distribution with the density function f ( s ) = s exp( - s/θ ) 2 , s > 0 . Deﬁne a region C k = { ( x 1 ,x 2 ) : x 1 + x 2 > k } for some positive k . Suppose the test with critical region C k is used to test H 0 : θ 1 against H 1 : θ > 1 . (a) Find the power function of the test and show that it is an increasing function in θ . (b) Deduce from (a) the size of the test. 2. Let X 1 ,...,X n be i.i.d. Poisson random variables with mean θ . (a) Find the most powerful size α test of H 0 : θ = 1 vs H 1 : θ = 1 . 21 (b) Use the Central Limit Theorem to show that the smallest value of n required to make α = 0 . 05 and β 0 . 1 ( where α and β are the type I and II error probabilities respectively) is somewhere near 212. 3. A random variable

This preview has intentionally blurred sections. Sign up to view the full version.

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

## This note was uploaded on 04/15/2011 for the course STAT 1302 taught by Professor Smslee during the Spring '10 term at HKU.

### Page1 / 3

class8 q - THE UNIVERSITY OF HONG KONG DEPARTMENT OF...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online