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Let \(X_{1}, X_{2}, \ldots X_{n}\) be a random sample of size \(n\) from a distribution with probability density function
\[ f(x, \theta) = \frac{1}{\theta}e^{-x/\theta}, \quad x > 0, \ \theta > 0 \]
Note that, the moments of this distribution are given by
\[ E[X^k] = \int_{0}^{\infty} \frac{x^k}{\theta}e^{-x/\theta} = k! \cdot \theta^k. \]
This will be a useful fact for Exercises 2 and 3.
(a) Obtain the maximum likelihood estimator of \(\theta\), \(\hat{\theta}\). (This should be a function of the unobserved \(x_i\) and the sample size \(n\).) Calculate the estimate when
\[ x_{1} = 0.50, \ x_{2} = 1.50, \ x_{3} = 4.00, \ x_{4} = 3.00. \]
(This should be a single number, for this dataset.)
(b) Calculate the bias of the maximum likelihood estimator of \(\theta\), \(\hat{\theta}\). (This will be a number.)
(c) Find the mean squared error of the maximum likelihood estimator of \(\theta\), \(\hat{\theta}\). (This will be an expression based on the parameter \(\theta\) and the sample size \(n\). Be aware of your answer to the previous part, as well as the distribution given.)
(d) Provide an estimate for \(P[X > 4]\) when
\[ x_{1} = 0.50, \ x_{2} = 1.50, \ x_{3} = 4.00, \ x_{4} = 3.00. \]
Let \(X_{1}, X_{2}, \ldots X_{n}\) be a random sample of size \(n\) from a distribution with probability density function
\[ f(x, \alpha) = \alpha^{-2}xe^{-x/\alpha}, \quad x > 0, \ \alpha > 0 \]
(a) Obtain the maximum likelihood estimator of \(\alpha\), \(\hat{\alpha}\). Calculate the estimate when
\[ x_{1} = 0.25, \ x_{2} = 0.75, \ x_{3} = 1.50, \ x_{4} = 2.5, \ x_{5} = 2.0. \]
(b) Obtain the method of moments estimator of \(\alpha\), \(\tilde{\alpha}\). Calculate the estimate when
\[ x_{1} = 0.25, \ x_{2} = 0.75, \ x_{3} = 1.50, \ x_{4} = 2.5, \ x_{5} = 2.0. \]
Let \(X_{1}, X_{2}, \ldots X_{n}\) be a random sample of size \(n\) from a distribution with probability density function
\[ f(x, \beta) = \frac{1}{2 \beta^3} x^2 e^{-x/\beta}, \quad x > 0, \ \beta > 0 \]
(a) Obtain the maximum likelihood estimator of \(\beta\), \(\hat{\beta}\). Calculate the estimate when
\[ x_{1} = 2.00, \ x_{2} = 4.00, \ x_{3} = 7.50, \ x_{4} = 3.00. \]
(b) Obtain the method of moments estimator of \(\beta\), \(\tilde{\beta}\). Calculate the estimate when
\[ x_{1} = 2.00, \ x_{2} = 4.00, \ x_{3} = 7.50, \ x_{4} = 3.00. \]
Let \(X_{1}, X_{2}, \ldots X_{n}\) be a random sample of size \(n\) from a distribution with probability density function
\[ f(x, \lambda) = \lambda x^{\lambda - 1}, \quad 0 < x < 1, \lambda > 0 \]
(a) Obtain the maximum likelihood estimator of \(\lambda\), \(\hat{\lambda}\). Calculate the estimate when
\[ x_{1} = 0.10, \ x_{2} = 0.20, \ x_{3} = 0.30, \ x_{4} = 0.40. \]
(b) Obtain the method of moments estimator of \(\lambda\), \(\tilde{\lambda}\). Calculate the estimate when
\[ x_{1} = 0.10, \ x_{2} = 0.20, \ x_{3} = 0.30, \ x_{4} = 0.40. \]