285 Problems 1. Let X1,.. .,X, be a sample from the distribution whose density function is f (x) = { 0. otherwise Determine the maximum likelihood estimator of 0. 2. Determine the maximum likelihood estimator of 0 when X1,...,X,n is a sample with density function f(x) = te-lx-0), 2 Let X1,..., X, be a sample from a normal l,o population. Determine the max- um likelihood estimator of o when u is known. What is the expected value of this estimator? 4 Determine the maximum likelihood estimates of a and A when X1,...,X, is a sample from the Pareto density function rax-(a+1), if x > a f(x) = 0, %3D if x 마 %3 .01 where D is the discharge of the largest flood in a randomly chosen year. The following table gives the flood discharges of the largest floods of the Blackstone River in Woonsocket, Rhode Island, in each of the years from 1929 to 1965. Assuming that these discharges follow a lognormal distribution, estimate the value of a 100-year flood.

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285
Problems
1. Let X1,.. .,X, be a sample from the distribution whose density function is
f (x) = {
0.
otherwise
Determine the maximum likelihood estimator of 0.
2. Determine the maximum likelihood estimator of 0 when X1,...,X,n is a sample
with density function
f(x) = te-lx-0),
2 Let X1,..., X, be a sample from a normal l,o population. Determine the max-
um likelihood estimator of o when u is known. What is the expected value
of this estimator?
4 Determine the maximum likelihood estimates of a and A when X1,...,X, is a
sample from the Pareto density function
rax-(a+1),
if x > a
f(x) =
0,
%3D
if x <a
5. Suppose that X1,..., Xn are normal with mean u1; Y1,..., Yn are normal with
mean u2; and W1,..., Wn are normal with mean u1 + u 2. Assuming that all 3n
random variables are independent with a common variance, find the maximum
likelihood estimators of u1 and u2.
6. River floods are often measured by their discharges (in units of feet cubed per
second). The value v is said to be the value of a 100-year flood if
P{D> 마 %3 .01
where D is the discharge of the largest flood in a randomly chosen year. The
following table gives the flood discharges of the largest floods of the Blackstone
River in Woonsocket, Rhode Island, in each of the years from 1929 to 1965.
Assuming that these discharges follow a lognormal distribution, estimate the value
of a 100-year flood.
Transcribed Image Text:285 Problems 1. Let X1,.. .,X, be a sample from the distribution whose density function is f (x) = { 0. otherwise Determine the maximum likelihood estimator of 0. 2. Determine the maximum likelihood estimator of 0 when X1,...,X,n is a sample with density function f(x) = te-lx-0), 2 Let X1,..., X, be a sample from a normal l,o population. Determine the max- um likelihood estimator of o when u is known. What is the expected value of this estimator? 4 Determine the maximum likelihood estimates of a and A when X1,...,X, is a sample from the Pareto density function rax-(a+1), if x > a f(x) = 0, %3D if x <a 5. Suppose that X1,..., Xn are normal with mean u1; Y1,..., Yn are normal with mean u2; and W1,..., Wn are normal with mean u1 + u 2. Assuming that all 3n random variables are independent with a common variance, find the maximum likelihood estimators of u1 and u2. 6. River floods are often measured by their discharges (in units of feet cubed per second). The value v is said to be the value of a 100-year flood if P{D> 마 %3 .01 where D is the discharge of the largest flood in a randomly chosen year. The following table gives the flood discharges of the largest floods of the Blackstone River in Woonsocket, Rhode Island, in each of the years from 1929 to 1965. Assuming that these discharges follow a lognormal distribution, estimate the value of a 100-year flood.
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