Let X1,., X, be a random sample from a uniform distribution on the interval (0, 50], where 0 > 0. Thus, the population pdf is if 0 < r< 50 50 fo(x) = 0, otherwise. (a) Show that the population mean and variance are: E,(X) = (2.5)0 and V,(X) = (登)P. (b) Derive a method of moments estimator of 0 based on X1,..., Xn. (c) Derive the bias and MSE formulas for your method of moments estimator. (d) Give the likelihood function, clearly specifying its argument and domain (input space). Draw a rough sketch of the likelihood function and derive the maximum likelihood estimator of 0, with clear explanation.
Let X1,., X, be a random sample from a uniform distribution on the interval (0, 50], where 0 > 0. Thus, the population pdf is if 0 < r< 50 50 fo(x) = 0, otherwise. (a) Show that the population mean and variance are: E,(X) = (2.5)0 and V,(X) = (登)P. (b) Derive a method of moments estimator of 0 based on X1,..., Xn. (c) Derive the bias and MSE formulas for your method of moments estimator. (d) Give the likelihood function, clearly specifying its argument and domain (input space). Draw a rough sketch of the likelihood function and derive the maximum likelihood estimator of 0, with clear explanation.
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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Could you please help solve this a)- e)
![Let X1, , X, be a random sample from a uniform distribution on the
interval (0, 50), where 0 > 0. Thus, the population pdf is
, if 0 < x < 50
50
fo(x) =
0,
otherwise.
(a) Show that the population mean and variance are: E,(X) = (2.5)0 and V(X) =
(b) Derive a method of moments estimator of 0 based on X1,..., Xn-
(c) Derive the bias and MSE formulas for your method of moments estimator.
(d) Give the likelihood function, clearly specifying its argument and domain (input
space). Draw a rough sketch of the likelihood function and derive the maximum likelihood
estimator of 0, with clear explanation.
(e) Let V = () max{X1,..., Xn}. Derive the cdf of V, that is, derive Fo(v) = Po[V <
%3D
v] for all -o < v < oo.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7157b572-b6ea-402e-940f-02951f488872%2F62ef6fce-92e7-41bc-bf3b-f11394a5b99a%2F93lb6n_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Let X1, , X, be a random sample from a uniform distribution on the
interval (0, 50), where 0 > 0. Thus, the population pdf is
, if 0 < x < 50
50
fo(x) =
0,
otherwise.
(a) Show that the population mean and variance are: E,(X) = (2.5)0 and V(X) =
(b) Derive a method of moments estimator of 0 based on X1,..., Xn-
(c) Derive the bias and MSE formulas for your method of moments estimator.
(d) Give the likelihood function, clearly specifying its argument and domain (input
space). Draw a rough sketch of the likelihood function and derive the maximum likelihood
estimator of 0, with clear explanation.
(e) Let V = () max{X1,..., Xn}. Derive the cdf of V, that is, derive Fo(v) = Po[V <
%3D
v] for all -o < v < oo.
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