a) It is known that the random variable Y has a probability density function @ye-1, 0

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A It Is Known That The Random Variable Y Has A Probability Density Function Ye 1 0 Y 1 Fly 0 Lo Elsewhere It Is D 1 (70.35 KiB) Viewed 90 times

a) It is known that the random variable Y has a probability density function @ye-1, 0<y<1 fly;0) = lo, elsewhere It is desired to test the null hypothesis H.: = 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 H, if y, s (4) 3 Page SSTA022 NOV/DEC EXAMINATION 2021 ii) Consider the test which rejects H, if y, sc. If the level of significance for the test is 0.1. What is the value of c? (4) iii) Show that the product T1=1 yi is sufficient statistic for using the factorisation method. (5) iv) Show that the product IT-17; is minimal sufficient for e. (6) b) Let Yı, Y2, ..., Y, 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 n Ülyiso i=1 where c solves the probability equation a = P(1= Y; Scle = 2). (8) Let X1, X2, ..., Xn be a random sample from GAMMA(2.b) distribution, and consider Y = !=1X;. c) Show whether or not Y is a pivotal quantity and give its distribution. (4)