Here are the problems written out, in case you don't have the text:

• 6.1: Let X_1, ..., X_n be a random sample from each of the distributions having the following probability density functions:
  (b) f(x; theta) = theta * x^(theta - 1), 0 < x < 1, 0 < theta < infinity, zero elsewhere
  (d) f(x; theta) = 1/2 * exp( - | x - theta | ), where |x| is the absolute value of x, and the ranges of x and theta are the real line
  In each case, find the mle theta-hat of theta.
• 6.7: For each of the distributions in exercise 6.1, find an estimator of theta by the method of moments and show that it is consistent (hint: to show consistency, consider using Slutsky's theorem and the fact that if a sequence converges in distribution to a constant, then it converges in probability to that constant) .
• 6.9: Let X_1, ..., X_n be iid each with a distribution with pdf f(x;theta) = (1/theta) exp( - x/theta ), 0 < x < infinity, zero elsewhere. Find the mle of P(X <= 2).
• 6.10: Let X have a binomial distribution with parameters n and p. The variance of X/n is p(1-p)/n; this is sometimes estimated by the mle X/n * (1 - X/n) / n. Is this an unbiased estimator of p(1-p)/n? If not, can you construct one by multiplying this one by a constant?
• 6.12: Let Y_1 < Y_2 < ... < Y_n be the order statistics of a random sample of size n from the uniform distribution of the continuous type over the closed interval [theta - rho, theta + rho]. Find the maximum likelihood estimators for theta and rho. Are these two unbiased estimators?