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Suppose that \(X_1, X_2, \ldots, X_n\) and \(Y_1, Y_2, \ldots, Y_n\) are independent random samples from populations with the same mean \(\mu\) and variances \(\sigma_X^2\) and \(\sigma_Y^2\), respectively.

That is,

\[ \begin{aligned} X_i &\sim \text{N}(\mu, \sigma^2_X) \\ Y_i &\sim \text{N}(\mu, \sigma^2_Y) \end{aligned} \]

Show that \(\displaystyle\frac{2\bar{X} + 3\bar{Y}}{5}\) is a consistent estimator of \(\mu\).

Let \(Y_1, Y_2, \ldots, Y_n\) denote a random sample from the probability density function

\[ f(y \mid \theta) = \theta y^{\theta-1}, \quad 0 < y < 1, \theta > 0. \]

Show that \(\bar{Y}\) is a consistent estimator of \(\displaystyle\frac{\theta}{\theta + 1}\).

Consider two binomial random variables \(Y_1\) and \(Y_2\). In particular,

\[ \begin{aligned} Y_1 &\sim \text{binom}(n, p_1) \\ Y_2 &\sim \text{binom}(n, p_2) \end{aligned} \]

Propose and justify a consistent estimator for \(p_1 - p_2\).

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

\[ x_{1} = 0.50, \ x_{2} = 0.75, \ x_{3} = 0.85, \ x_{4} = 0.25. \]

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 mean squared error of this estimator when estimating \(\lambda\). (Your answer will be a function of the sample size \(n\) and \(\lambda\).)