# Exercise 1

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

$x_{1} = 0.50, \ x_{2} = 0.75, \ x_{3} = 0.80, \ x_{4} = 0.25.$

# Exercise 2

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 variance of this estimator. (Your answer will be a function of the sample size $$n$$ and $$\lambda$$.)

# Exercise 3

Let $$X_{1}, X_{2}, \ldots X_{n}$$ be a random sample of size $$n$$ from a distribution with probability density function

$f(x \mid \lambda) = \lambda x^{\lambda - 1}, \quad 0 < x < 1, \ \lambda > 0$

Obtain the maximum likelihood estimator of $$\lambda$$, $$\hat{\lambda}$$. Calculate an estimate using this maximum likelihood estimator when

$x_{1} = 0.10, \ x_{2} = 0.20, \ x_{3} = 0.30, \ x_{4} = 0.40.$

# Exercise 4

Let $$X_1, X_2, \ldots, X_n$$ denote independent and identically distributed uniform random variables on the interval $$[0, 3\beta]$$.

Obtain the maximum likelihood estimator for $$\beta$$, $$\hat{\beta}$$. Use this estimator to provide an estimate of $$\text{Var}[X]$$ when

$x_{1} = 1.3, \ x_{2} = 5.7, \ x_{3} = 2.2.$

# Exercise 5

Let $$X_{1}, X_{2}, \ldots X_{n}$$ be a random sample of size $$n$$ from a distribution with probability density function

$f(x \mid \theta) = \frac{1}{\theta}e^{-x/\theta}, \quad x > 0, \ \theta > 0$

Obtain the maximum likelihood estimator of $$\theta$$, $$\hat{\theta}$$. Use this maximum likelihood estimator to obtain an estimate of

$P[X > 4]$

when

$x_{1} = 0.50, \ x_{2} = 1.50, \ x_{3} = 4.00, \ x_{4} = 3.00.$