1 (X₁, To test a proportion of defective items one draws a sample X₁ = population with an unknown € (0, 1). Give the likelihood function of the sample. (ii) Find the m.l.e. and MLE of 0. Hint. (i) We can write the density f(x|0) of each r.v. X₁ as x = 0,1 0² (1 - 0)¹-²1(0,1) (0), 0, otherwise. f (x 0 ) = {' Thus, the likelihood function is fn(Xn|0) = 0° (1 –0)”°1(o,n)(0) where 0 : = x₁ + ... + än. (Explain why.) (ii) The value of that maximizes the likelihood function is the same value that maximizes the log of fn (Xn0) since the log function is strictly monotone. So, let L(0) = lnfn (xn|0) = oln0 + (n − σ)ln(1 – 0). - Then, show that Xn) from a Bernoulli L'(0) = n(1-0),
1 (X₁, To test a proportion of defective items one draws a sample X₁ = population with an unknown € (0, 1). Give the likelihood function of the sample. (ii) Find the m.l.e. and MLE of 0. Hint. (i) We can write the density f(x|0) of each r.v. X₁ as x = 0,1 0² (1 - 0)¹-²1(0,1) (0), 0, otherwise. f (x 0 ) = {' Thus, the likelihood function is fn(Xn|0) = 0° (1 –0)”°1(o,n)(0) where 0 : = x₁ + ... + än. (Explain why.) (ii) The value of that maximizes the likelihood function is the same value that maximizes the log of fn (Xn0) since the log function is strictly monotone. So, let L(0) = lnfn (xn|0) = oln0 + (n − σ)ln(1 – 0). - Then, show that Xn) from a Bernoulli L'(0) = n(1-0),
MATLAB: An Introduction with Applications
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Author:Amos Gilat
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Chapter1: Starting With Matlab
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Question
Please look at the above images, which include the question and a process of how to solve the question. Please explain and show work for the parts above where it says "explain" to get the desired answer.
![1
(X₁,
To test a proportion of defective items one draws a sample X₁ =
population with an unknown € (0, 1).
Give the likelihood function of the sample.
(ii) Find the m.l.e. and MLE of 0.
Hint. (i) We can write the density f(x|0) of each r.v. X₁ as
x = 0,1
0² (1 - 0)¹-²1(0,1) (0),
0,
otherwise.
f (x 0 ) = {'
Thus, the likelihood function is
fn(Xn|0) = 0° (1 –0)”°1(o,n)(0)
where
0 : = x₁ + ... + än. (Explain why.)
(ii) The value of that maximizes the likelihood function is the same value that maximizes the log of
fn (Xn0) since the log function is strictly monotone. So, let
L(0) = lnfn (xn|0) = oln0 + (n − σ)ln(1 – 0).
-
Then, show that
Xn) from a Bernoulli
L'(0) = n(1-0),](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdb82eee3-c597-43b7-b98f-cffc18fbca72%2Ffc58de1a-98f1-4999-bc0c-bce5a17a0325%2Fa04upl_processed.png&w=3840&q=75)
Transcribed Image Text:1
(X₁,
To test a proportion of defective items one draws a sample X₁ =
population with an unknown € (0, 1).
Give the likelihood function of the sample.
(ii) Find the m.l.e. and MLE of 0.
Hint. (i) We can write the density f(x|0) of each r.v. X₁ as
x = 0,1
0² (1 - 0)¹-²1(0,1) (0),
0,
otherwise.
f (x 0 ) = {'
Thus, the likelihood function is
fn(Xn|0) = 0° (1 –0)”°1(o,n)(0)
where
0 : = x₁ + ... + än. (Explain why.)
(ii) The value of that maximizes the likelihood function is the same value that maximizes the log of
fn (Xn0) since the log function is strictly monotone. So, let
L(0) = lnfn (xn|0) = oln0 + (n − σ)ln(1 – 0).
-
Then, show that
Xn) from a Bernoulli
L'(0) = n(1-0),
![where is the sample mean of x₁, ..., n.
It is easily seen that L'(0) changes its sign from positive to negative by passing through an which of
course is a value from [0, 1]. Thus, the empirical sample mean is the m.l.e. of 0.
[Notice that a more elaborate analysis in which we distinguish the cases with
other values of n from (0, 1) yields the same result. ]
Thus, ê=X₂ is the MLE of the sample.
1
n = 0 or n = n from all](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdb82eee3-c597-43b7-b98f-cffc18fbca72%2Ffc58de1a-98f1-4999-bc0c-bce5a17a0325%2Fq0ju1e9_processed.png&w=3840&q=75)
Transcribed Image Text:where is the sample mean of x₁, ..., n.
It is easily seen that L'(0) changes its sign from positive to negative by passing through an which of
course is a value from [0, 1]. Thus, the empirical sample mean is the m.l.e. of 0.
[Notice that a more elaborate analysis in which we distinguish the cases with
other values of n from (0, 1) yields the same result. ]
Thus, ê=X₂ is the MLE of the sample.
1
n = 0 or n = n from all
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