# Exercise 1

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 $$\theta$$.

# Exercise 2

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 $$\theta$$.

# Exercise 3

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 $$\mu^2$$.

# Exercise 4

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 $$\mu_X - \mu_Y$$.

# Exercise 5

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 $$\text{Var}[W] = 2k$$.

# Exercise 6

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 $$\lambda$$.

# Exercise 7

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 $$\theta$$.

# Exercise 8

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 $$\mu$$.

# Exercise 9

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 $$\beta$$.

# Exercise 10

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 $$\beta$$.