1. Let X1, X2, ..., X_n constitute a random sample from probability density function
given by
$$f(x; \theta) = \begin{cases}
\frac{2}{\theta^2}(\theta - x), & 0 \le x \le \theta \\
0, & \text{elsewhere}
\end{cases}$$
Find an estimator for θ by using method of moments.
2. Suppose that n independent Bernoulli trials, each of which is a success with
probability p, are performed. What is the maximum likelihood estimator of p?
3. Let X1, X2 ..., X_n denote a random sample of size n from this distribution whose
density is given by
$$f(x; \theta) = \theta x^{\theta - 1}, 0 < x < 1$$
Find the MME and the MLE of θ.
4. Find MMEs and MLEs of θ based on a random sample X1,X2..., Xn from each
of the following pdfs
a) f(x; θ) = (θ+1)x^-θ-2; 1 < x
b) f(x; θ) = θ²xe^-θx; 0 < x
5. If X1, X2..., X_n constitute a random sample of size n from a geometric population,
find formulas for estimating its parameter θ by using (a) the method of
moments; (b) the method of maximum likelihood.
6. Let X1, X2, ..., Xn constitute a random sample from probability density function
given by
$$f(x; \alpha, \theta) = \begin{cases}
\frac{1}{\Gamma(\alpha)\theta^\alpha}x^{\alpha - 1}e^{-\frac{x}{\theta}}, & x > 0 \\
0, & \text{elsewhere}
\end{cases}$$
Where α > 0 is known.
a. Find the MLE of θ.
b. Find the expected value and variance of θ.
