1 To test a proportion of defective items one draws a sample Xn population with an unknown 0 € (0, 1). (₂) Give the likelihood function of the sample. (22) Find the m.l.e. and MLE of 0. Hint. (2) We can write the density f(x|0) of each r.v. X as 0² (10)¹-1(0.1) (0), ƒ (x|0) = {' {0² (1. 0, x = 0,1 otherwise. Thus, the likelihood function is fn (₂0) = 0° (1 - 0) 1 (0,1) (0) where = L' (0) = n(i=8); where is the sample mean of x₁,..., n. (X₁,..., Xn) from a Bernoulli 0 = = 2₁+ ... +än. (Explain why.) (22) The value of that maximizes the likelihood function is the same value that maximizes the log of fn (n) since the log function is strictly monotone. So, let L(0) : = Infn (xn|0) = oln0 + (n − o)ln(1 — 0). Then, show that
1 To test a proportion of defective items one draws a sample Xn population with an unknown 0 € (0, 1). (₂) Give the likelihood function of the sample. (22) Find the m.l.e. and MLE of 0. Hint. (2) We can write the density f(x|0) of each r.v. X as 0² (10)¹-1(0.1) (0), ƒ (x|0) = {' {0² (1. 0, x = 0,1 otherwise. Thus, the likelihood function is fn (₂0) = 0° (1 - 0) 1 (0,1) (0) where = L' (0) = n(i=8); where is the sample mean of x₁,..., n. (X₁,..., Xn) from a Bernoulli 0 = = 2₁+ ... +än. (Explain why.) (22) The value of that maximizes the likelihood function is the same value that maximizes the log of fn (n) since the log function is strictly monotone. So, let L(0) : = Infn (xn|0) = oln0 + (n − o)ln(1 — 0). Then, show that
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
Related questions
Question
Please show all work when answering this question.
The question is at the very top of the first image and the hint that follows should help with the solution.
![It is easily seen that L'(0) changes its sign from positive to negative by passing through which of
course is a value from [0, 1]. Thus, the empirical sample mean ō is the m.l.e. Ĵ of 0.
[Notice that a more elaborate analysis in which we distinguish the cases with ān
other values of from (0, 1) yields the same result. ]
Thus, X, is the MLE of the sample.
72
=
1
=
0 orann from all](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdb82eee3-c597-43b7-b98f-cffc18fbca72%2F4c55efb2-dc1e-4bf0-898b-11ebd4f95960%2Ffe5cx7i_processed.png&w=3840&q=75)
Transcribed Image Text:It is easily seen that L'(0) changes its sign from positive to negative by passing through which of
course is a value from [0, 1]. Thus, the empirical sample mean ō is the m.l.e. Ĵ of 0.
[Notice that a more elaborate analysis in which we distinguish the cases with ān
other values of from (0, 1) yields the same result. ]
Thus, X, is the MLE of the sample.
72
=
1
=
0 orann from all
Transcribed Image Text:1
To test a proportion of defective items one draws a sample Xn
population with an unknown 0 € (0, 1).
(₂) Give the likelihood function of the sample.
(22) Find the m.l.e. and MLE of 0.
Hint. (2) We can write the density f(x|0) of each r.v. X as
0² (10)¹-1(0.1) (0),
ƒ (x|0) = {'
{0² (1. 0,
x = 0,1
otherwise.
Thus, the likelihood function is
fn (₂0) = 0° (1 - 0) 1 (0,1) (0)
where
=
L' (0) = n(i=8);
where is the sample mean of x₁,..., n.
(X₁,..., Xn) from a Bernoulli
0 = = 2₁+ ... +än. (Explain why.)
(22) The value of that maximizes the likelihood function is the same value that maximizes the log of
fn (n) since the log function is strictly monotone. So, let
L(0) : = Infn (xn|0) = oln0 + (n − o)ln(1 — 0).
Then, show that
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