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 f(x; 0) = [ (0 + 1)x® where -1 < 8. A random sample of ten students yields data x₁ = 0.86, x₂ = 0.92, x3 = 0.95, x4 = 0.94, X5 = 0.45, x6 = 0.65, x = 0.73, xg = 0.90, x9 = 0.79, X10 = 0.79. (a) Use the method of moments to obtain an estimator of Ⓒ 1 (1 + X) 1 0 (x - 1) - 0 (1 + 7) - 1 1 0 (1-7) - - 2 -n ΣIn(X) Eln(X;) n 2 1 0 (7-1) - Compute the estimate for this data. (Round your answer to two decimal places.) 0≤x≤ 1 otherwise 1 (b) Obtain the maximum likelihood estimator of 0. Eln(X;) n ΣIn(X;) -n - 1 1 n 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 f(x; 0) = [ (0 + 1)x® where -1 < 8. A random sample of ten students yields data x₁ = 0.86, x₂ = 0.92, x3 = 0.95, x4 = 0.94, X5 = 0.45, x6 = 0.65, x = 0.73, xg = 0.90, x9 = 0.79, X10 = 0.79. (a) Use the method of moments to obtain an estimator of Ⓒ 1 (1 + X) 1 0 (x - 1) - 0 (1 + 7) - 1 1 0 (1-7) - - 2 -n ΣIn(X) Eln(X;) n 2 1 0 (7-1) - Compute the estimate for this data. (Round your answer to two decimal places.) 0≤x≤ 1 otherwise 1 (b) Obtain the maximum likelihood estimator of 0. Eln(X;) n ΣIn(X;) -n - 1 1 n ΣIn(X;) Compute the estimate for the given 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
f(x; 0) =
(0+1)x
where -1 < 0. A random sample of ten students yields data x₁ = 0.86, x₂ = 0.92, x3 = 0.95, x4 = 0.94, X5 = 0.45, x6 = 0.65,
x = 0.73, xg = 0.90, x9 = 0.79, X10 = 0.79.
(a) Use the method of moments to obtain an estimator of Ⓒ
1
(1 + X)
- 1
1
0 (x - 1) -
- 2
0 (1-7) -
-n
ΣIn(X)
Eln(X;)
n
2
0 (1 + 7)
1
0 (7-1) -
Compute the estimate for this data. (Round your answer to two decimal places.)
0≤x≤ 1
otherwise
1
(b) Obtain the maximum likelihood estimator of 0.
Eln(X;)
n
ΣΙn(Χ;)
-n
- 1
1
n
ΣIn(Χ.)
Compute the estimate for the given data. (Round your answer to two decimal places.)
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