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) (6) X, X. X, form a random sample from a distribution whose PDF is Ar, 0) = 10. otherwise, where the value of the positive parameter 0 is unknown. Determine the MLE of the median of the distribution. (ii) There is widespread agreement amongst the managers of the Reliable Motor Company that the number x of faulty cars produced in a month has a binomial distribution P(x=s) =| p** (s=0, 1. ...."; 0sp

) (6) X, X. X, form a random sample from a distribution whose PDF is Ar, 0) = 10. otherwise, where the value of the positive parameter 0 is unknown. Determine the MLE of the median of the distribution. (ii) There is widespread agreement amongst the managers of the Reliable Motor Company that the number x of faulty cars produced in a month has a binomial distribution P(x=s) =| p** (s=0, 1. ...."; 0sp

MATLAB: An Introduction with Applications
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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) (i) X,, X2, ..., X, form a random sample from a distribution whose PDF is | 2x/0* , 0 < xSI S(r, 6) = |0, otherwise, where the value of the positive parameter 0 is unknown. Determine the MLE of the median of the distribution. (ii) There is widespread agreement amongst the managers of the Reliable Motor Company that the number x of faulty cars produced in a month has a binomial distribution P(x=s)=| p)*** (s =0, 1, .... n: 0<p<l). There is, however, some dispute about the parameter p. The general manager has a prior distribution for p which is uniform (i.e. with the PDF f,(x) = /(0 < x< 1)), while the more pessimistic production manager has a prior distribution with density f,(x)=2x/(0< IS1). Both PDFS are concentrated on (0, 1). In a particular month, s faulty cars are produced. Show that if the general manager's loss function is (p – py’, where p is her estimate and p is the true value, then her best estimate of p is s+1 n+2 The production manager has responsibilities different from those of the general man- ager, and a different loss function given by (1 – p)(p – p)². Find his best estimator of p and show that it is greater than that of the general manager unless s2n/2. You may assume that, for non-negative integers a, ß. a!B! (a +B + 1)!
Transcribed Image Text:) (i) X,, X2, ..., X, form a random sample from a distribution whose PDF is | 2x/0* , 0 < xSI S(r, 6) = |0, otherwise, where the value of the positive parameter 0 is unknown. Determine the MLE of the median of the distribution. (ii) There is widespread agreement amongst the managers of the Reliable Motor Company that the number x of faulty cars produced in a month has a binomial distribution P(x=s)=| p)*** (s =0, 1, .... n: 0<p<l). There is, however, some dispute about the parameter p. The general manager has a prior distribution for p which is uniform (i.e. with the PDF f,(x) = /(0 < x< 1)), while the more pessimistic production manager has a prior distribution with density f,(x)=2x/(0< IS1). Both PDFS are concentrated on (0, 1). In a particular month, s faulty cars are produced. Show that if the general manager's loss function is (p – py’, where p is her estimate and p is the true value, then her best estimate of p is s+1 n+2 The production manager has responsibilities different from those of the general man- ager, and a different loss function given by (1 – p)(p – p)². Find his best estimator of p and show that it is greater than that of the general manager unless s2n/2. You may assume that, for non-negative integers a, ß. a!B! (a +B + 1)!
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