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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 \(p_1 - p_2\).

If \(Y_1, Y_2, \ldots, Y_n\) denote a random sample from a geometric distribution with parameter \(p\), show that \(\bar{Y}\) is sufficient for \(p\).

If \(X_1, X_2, \ldots, X_n\) denote a random sample from the probability density function

\[ f(x \mid \beta) = \frac{1}{2\beta^3} x^2 e^{-x/\beta}, \quad x > 0, \beta > 0. \]

Show that \(\bar{X}\) is sufficient for \(\beta\).