 # (1) Suppose that Y 1 , ,Y n n is a random sample where the dens...

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(1) Suppose that Y 1 , ,Y n n is a random sample where the density of each random variable Y is fy (y)=202y-3, y 0 0 for some parameter 0>1. Let 0 III min {Y1 Yn}. (a) Calculate B(ô), the bias of 0. (b) Determine the value of the constant c for which 0 =cô is unbiased. (c) Calculate the mean-square error of 01. (2) Suppose that are independent and identically distributed Exp(8) random variables. (a) Let X1=Y(+...+Y,.Use moment generating functions to show that X 1 has a Gamma(n,6 distribution. (b) Let that X 2 has an Exp of distribution. n (c) Verify that III is an unbiased estimator for 0. (d) Verify that 0 =nX2 is an unbiased estimator for 0. (e) Which estimator, 0 or O2 is preferred for the estimation of O? (4) Suppose that Y is normally distributed with mean 0 and unknown variance of Y 2 (a) Show that the quantity is a pivotal quantity. 2 (b) Use the pivotal quantity 3/6 to construct a 95% confidence interval for of y² (5) Let Y1,Y2, and Y3 be a random sample from a normal distribution with mean u and variance o2 , where both u and 62 are unknown. Consider the following estimators for u: = - 1 + - 2 + - 3 4 2 4 1 1 1 A2 = - 3 Y1 + - 3 Y2 + - 3 Y3 (a) Show that A1 and A2 are unbiased estimators for u. (b) Find the variances of A and A2. (c) Find the efficiency of A2 relative to A1. , and tell me which is more efficient estimator for u. (6) Let Y1, , Y2 Yn n denote a random sample from the probability density function 0 y8-1 , 00 f(y/0)= , elsewhere (a) Show that On =Y is unbiased estimator for 0 0+1 (b) Find the variance of On. (c) Show that On is a consistent estimator for = 0 0+1

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