# Exercise 1

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 $$\mu$$.

# Exercise 2

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 $$\displaystyle\frac{\theta}{\theta + 1}$$.

# Exercise 3

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 $$p_1 - p_2$$.

# Exercise 4

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 $$p$$.

# Exercise 5

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 $$\beta$$.