## Transcribed Text

(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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