Let X denote the proportion of allotted time that a randomly selected student spends working on a certain aptitude test. Suppose the pdf of X is |(0 + 1)xº 0sxs1 f(x; 0) = • otherwise where -1 < 0. A random sample of ten students yields data x, = 0.99, x2 = 0.45, x3 = 0.94, x4 = 0.92, x5 = 0.79, x6 = 0.86, x, = 0.73, xg = 0.90, x9 = 0.79, x10 = 0.65. (a) Use the method of moments to obtain an estimator of e 1. 1+ 1 - 2 (1 - X - 1 + - 2 X- 1 (R -1) - 1 Compute the estimate for this data. (Round your answer to two decimal places.)

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(b) Obtain the maximum likelihood estimator of 0.
Eln(X;)
in
-n
1
Σin(X)
Σin(X)
-n
Σin(X)
in
Σin(X)
Compute the estimate for the given data. (Round your answer to two decimal places.)
Transcribed Image Text:(b) Obtain the maximum likelihood estimator of 0. Eln(X;) in -n 1 Σin(X) Σin(X) -n Σin(X) in Σin(X) Compute the estimate for the given data. (Round your answer to two decimal places.)
Let X denote the proportion of allotted time that a randomly selected student spends working on a certain aptitude test. Suppose the pdf of X is
(0 + 1)xº
0 < x < 1
f(x; 0)
otherwise
where -1 < 0. A random sample of ten students yields data x,
0.99, x2 = 0.45, x3 = 0.94, x4 = 0.92, x5 = 0.79, x6 = 0.86, X7
0.73, xg = 0.90, X9 = 0.79, x10 = 0.65.
%3D
(a) Use the method of moments to obtain an estimator of 0
1
1 + X
1
- 2
1 - X
1
- 1
1 + X
X - 1
1
- 1
1
Compute the estimate for this data. (Round your answer to two decimal places.)
Transcribed Image Text:Let X denote the proportion of allotted time that a randomly selected student spends working on a certain aptitude test. Suppose the pdf of X is (0 + 1)xº 0 < x < 1 f(x; 0) otherwise where -1 < 0. A random sample of ten students yields data x, 0.99, x2 = 0.45, x3 = 0.94, x4 = 0.92, x5 = 0.79, x6 = 0.86, X7 0.73, xg = 0.90, X9 = 0.79, x10 = 0.65. %3D (a) Use the method of moments to obtain an estimator of 0 1 1 + X 1 - 2 1 - X 1 - 1 1 + X X - 1 1 - 1 1 Compute the estimate for this data. (Round your answer to two decimal places.)
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