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Let \(X_1, X_2, \ldots, X_n\) be a random sample from a distribution with probability density function
\[ f(x \mid \theta) = \left(\theta^2 + \theta\right)x^{\theta - 1}(1 - x), \quad 0 < x < 1, \theta > 0. \]
Obtain a method of moments estimator for \(\theta\), \(\tilde{\theta}\). Calculate an estimate using this estimator when
\[ x_{1} = 0.50, \ x_{2} = 0.75, \ x_{3} = 0.80, \ x_{4} = 0.25. \]
Let \(Y_1, Y_2, \ldots, Y_n\) denote independent and identically distributed uniform random variables on the interval \((0, 4\lambda)\).
Obtain a method of moments estimator for \(\lambda\), \(\tilde{\lambda}\). Calculate the variance of this estimator. (Your answer will be a function of the sample size \(n\) and \(\lambda\).)
Let \(X_{1}, X_{2}, \ldots X_{n}\) be a random sample of size \(n\) from a distribution with probability density function
\[ f(x \mid \lambda) = \lambda x^{\lambda - 1}, \quad 0 < x < 1, \ \lambda > 0 \]
Obtain the maximum likelihood estimator of \(\lambda\), \(\hat{\lambda}\). Calculate an estimate using this maximum likelihood estimator when
\[ x_{1} = 0.10, \ x_{2} = 0.20, \ x_{3} = 0.30, \ x_{4} = 0.40. \]
Let \(X_1, X_2, \ldots, X_n\) denote independent and identically distributed uniform random variables on the interval \([0, 3\beta]\).
Obtain the maximum likelihood estimator for \(\beta\), \(\hat{\beta}\). Use this estimator to provide an estimate of \(\text{Var}[X]\) when
\[ x_{1} = 1.3, \ x_{2} = 5.7, \ x_{3} = 2.2. \]
Let \(X_{1}, X_{2}, \ldots X_{n}\) be a random sample of size \(n\) from a distribution with probability density function
\[ f(x \mid \theta) = \frac{1}{\theta}e^{-x/\theta}, \quad x > 0, \ \theta > 0 \]
Obtain the maximum likelihood estimator of \(\theta\), \(\hat{\theta}\). Use this maximum likelihood estimator to obtain an estimate of
\[ P[X > 4] \]
when
\[ x_{1} = 0.50, \ x_{2} = 1.50, \ x_{3} = 4.00, \ x_{4} = 3.00. \]