Let \(X_1, X_2, \ldots, X_n\) be iid \(N(\theta,1)\) and consider \(\bar{X}_n = \frac{1}{n} \sum_{i=1}^n X_i\). Show that \(\bar{X}_n\) is a consistent estimator of \(\theta\).

Suppose that \(X_1, X_2, \ldots, X_n\) are an iid sample from the distribution

\[ f(x; \theta) = \frac{1}{2}(1+\theta x), \quad -1 < x < 1, -1 < \theta < 1. \]

Show that \(3 \bar{X}_n\) is a consistent estimator of \(\theta\).

Let \(Y_1, Y_2, \ldots, Y_n\) be a random sample such that

- \(\text{E}[Y_i] = \mu\)
- \(\text{Var}[Y_i] = \sigma^2\).

Suggest a consistent estimator for \(\mu^2\).

Let \(X_1, X_2, \ldots, X_n\) be iid \(N(\mu_X, \sigma^2_X\)). Also, let \(Y_1, Y_2, \ldots, Y_n\) be iid \(N(\mu_Y, \sigma^2_Y\)).

Suggest a consistent estimator for \(\mu_X - \mu_Y\).

Let \(X_1, X_2, \ldots, X_n\) be iid \(N(\mu_X, \sigma^2\)). Also, let \(Y_1, Y_2, \ldots, Y_n\) be iid \(N(\mu_Y, \sigma^2\)). Note that both distributions have the same variance.

Show that

\[ \frac{\sum_{i = 1}^{n}\left(X_i - \bar{X}\right)^2 + \sum_{i = 1}^{n}\left(Y_i - \bar{Y}\right)^2}{2n - 2} \]

is a consistent estimator for \(\sigma^2\).

Hint: Note that

\[ \frac{\sum_{i = 1}^{n}\left(X_i - \bar{X}\right)^2}{\sigma^2} \sim \chi^2_{n - 1} \]

\[ \frac{\sum_{i = 1}^{n}\left(Y_i - \bar{Y}\right)^2}{\sigma^2} \sim \chi^2_{n - 1} \]

Also, recall that, if \(W \sim \chi^2_k\), then \(\text{E}[W] = k\) and \(\text{Var}[W] = 2k\).

Let \(Y_1, Y_2, \ldots, Y_n\) be iid observations from a Poisson distribution with parameter \(\lambda\). Show that \(U = \sum_{i=1}^n Y_i\) is sufficient for \(\lambda\).

Let \(X_1, X_2, \ldots, X_n\) be iid observations from a distribution with density

\[ f(x \mid \theta) = \frac{\theta}{(1+x)^{\theta + 1}}, \quad 0< \theta < \infty, 0< x <\infty \]

Find a sufficient statistic for \(\theta\).

Let \(X_1, X_2, \ldots, X_n\) be iid observations from a normal distribution with a unknown mean, \(\mu\), and known variance \(\sigma^2 = 9\).

Show that \(\sum_{i = 1}^{n}X_i\) is a sufficient statistic for \(\mu\), then use this statistic to create an estimator that is both unbiased and sufficient for estimating \(\mu\).

Let \(Y_1, Y_2, \ldots, Y_n\) be iid observations from a distribution with density

\[ f(y \mid \beta) = \frac{y}{\beta}\cdot\exp\left(\frac{-y^2}{2\beta}\right), \quad y \geq 0, \beta > 0 \]

Find a sufficient statistic for \(\beta\).

Let \(X_1, X_2, \ldots, X_n\) be iid observations from a distribution with density

\[ f(x \mid \alpha, \beta) = \frac{\alpha}{\beta}\left(\frac{x}{\beta}\right)^{\alpha - 1}e^{-(x/\beta)^\alpha}, \quad x \geq 0, \alpha > 0, \beta > 0 \]

Let \(\alpha\) be a known constant and \(\beta\) be unknown. Find a sufficient statistic for \(\beta\).