Question

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; \theta) = \begin{cases} (\theta + 1)x^{\theta} & 0 \le x \le 1\\ 0 & \text{otherwise} \end{cases} where -1 < \theta. A random sample of ten students yields data x? = 0.94, x? = 0.90, x? = 0.79, x? = 0.92, x? = 0.95, x? = 0.73, x? = 0.65, x? = 0.49, x? = 0.79, x?? = 0.86. (a) Use the method of moments to obtain an estimator of \theta $\frac{1}{\bar{X} - 1} - 2$ $\frac{1}{1 + \bar{X}}$ $\frac{1}{1 - \bar{X}} - 2$ $\frac{1}{1 + \bar{X}} - 1$ $\frac{1}{\bar{X} - 1} - 1$ Compute the estimate for this data. (Round your answer to two decimal places.) (b) Obtain the maximum likelihood estimator of \theta. $\frac{\sum \ln(x_i)}{-n}$ $\frac{\sum \ln(x_i)}{n}$ $\frac{\sum \ln(x_i)}{n} - 1$ $\frac{\sum \ln(x_i)}{n}$ $\frac{-n}{\sum \ln(x_i)} - 1$ 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; \theta) = \begin{cases} (\theta + 1)x^{\theta} & 0 \le x \le 1\\ 0 & \text{otherwise} \end{cases}
where -1 < \theta. A random sample of ten students yields data x? = 0.94, x? = 0.90, x? = 0.79, x? = 0.92, x? = 0.95, x? = 0.73, x? = 0.65, x? = 0.49, x? = 0.79, x?? = 0.86.
(a) Use the method of moments to obtain an estimator of \theta
$\frac{1}{\bar{X} - 1} - 2$
$\frac{1}{1 + \bar{X}}$
$\frac{1}{1 - \bar{X}} - 2$
$\frac{1}{1 + \bar{X}} - 1$
$\frac{1}{\bar{X} - 1} - 1$
Compute the estimate for this data. (Round your answer to two decimal places.)
(b) Obtain the maximum likelihood estimator of \theta.
$\frac{\sum \ln(x_i)}{-n}$
$\frac{\sum \ln(x_i)}{n}$
$\frac{\sum \ln(x_i)}{n} - 1$
$\frac{\sum \ln(x_i)}{n}$
$\frac{-n}{\sum \ln(x_i)} - 1$
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; θ) = (θ+ 1)x^θ 0 ≤x ≤1
0 otherwise
where -1 < θ. A random sample of ten students yields data x? = 0.94, x? = 0.90, x? = 0.79, x? = 0.92, x? = 0.95, x? = 0.73, x? = 0.65, x? = 0.49, x? = 0.79, x?? = 0.86.
(a) Use the method of moments to obtain an estimator of θ(1)/(X̅ - 1) - 2
(1)/(1 + X̅)
(1)/(1 - X̅) - 2
(1)/(1 + X̅) - 1
(1)/(X̅ - 1) - 1
Compute the estimate for this data. (Round your answer to two decimal places.)
(b) Obtain the maximum likelihood estimator of θ.
(∑ln(xi))/(-n)
(∑ln(xi))/(n)
(∑ln(xi))/(n) - 1
(∑ln(xi))/(n)
(-n)/(∑ln(xi)) - 1
Compute the estimate for the given data. (Round your answer to two decimal places.)

Added by Enrique M.

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