1.1
Let X1, X2, ..., Xn denote a random sample from a Bernoulli distribution, X; ~B(1,p)
with pmf given by
$$f(x, p) = p^{x}(1-p)^{1-x}, x=0,1$$
i) Using the factorization theorem show that $\sum_{i=1}^{n} X_i$ is sufficient for p.
ii) Give the MVUE for p.
iii) Determine the variance of the MVUE
iv) Using the Cramer Rao Inequality find an efficient estimator for p.
1.2
Let X1, X2, ..., Xn denote a random sample from a geometric distribution with pmf,
$$f(x; p) = p(1-p)^{x-1}, x = 1,2, ...$$
i) Using this pmf, find the MLE for p.
ii) Find the MLE for E(X) and Var(X).
1.3
Let X1, X2, ..., Xn be a random sample from the distribution with pdf
$$f(x; \theta) = \frac{3}{\theta^3}x^2, 0 < x < \theta \text{ and } \theta > 0$$
Determine the MME for the unknown parameter $\theta$.
1.4
Let X1, X2, ..., Xn denote a random sample from a Normal distribution where X; ~N
(μ, σ²).
i) Find on the method of moment estimators for μ and σ²
ii) Find on the maximum likelihood estimators for μ and σ²
iii) Show that $\sum_{i=1}^{n} X_i$ and $\sum_{i=1}^{n} X_i^2$ are jointly, the minimal sufficient statistic for
parameters μ and σ²
iv) Use the Cramer-Rao Inequality to show that $\bar{X}$ is an efficient estimator of the mean
μ of a normal population when variance is known.
2.1
Let Y1, Y2,......, Y, denote a random sample from an exponential density function given
by
$$f(y; \theta) = (\frac{1}{\theta})e^{-y/\theta}, y > 0$$
i) Find a sufficient statistic for θ
ii) Find a MVUE for θ.
iii) Using (ii) above, find a MVUE for Var(Y).
2.2 Let X1, X2, ..., Xn represent a random sample having the following probability density function:
$$f(x; \theta) = (\theta + 1)x^{-\theta - 2}, x > 1, \text{ zero elsewhere.}$$
Find:
(i) the method of moments estimate (MME) and
(ii) the maximum likelihood estimate (MLE) for θ.
