STAT 311: Mathematical Statistics Homework 02 Chapter 5: Point Estimation 1- Let X, ,...,X, be a random sample of size n from the geometric distribution for which p is the probability of success. (a) Use the method of moments to find a point estimator for p. (b) Use the following data (simulated from geometric distribution) to find the moment estimator for p: 5 43 18 19 16 11 22 4 34 19 21 23 6. 21 12 2- The probability density of a one-parameter Weibull distribution is given by S2ax e¬ax² 0, x > 0 otherwise f(x) = - (a) Using a random sample of size n, obtain a moment estimator for a. (b) Assuming that the following data are from a one-parameter Weibull population, 1.87 1.60 2.36 1.12 0.15 1.83 0.64 1.53 0.73 2.26 3- Suppose X, ,...,X, are a random sample from an exponential distribution with parameter 0. Find the MLE of Ô. Also using the invariance property, obtain an MLE for the variance. 4- Let X1 .,X, be a random sample from a two-parameter Weibull distribution with pdf .... xa-1e(x/ß)“, x >0 f(x) 0, otherwise Find the MLES of a and ß. 5- Consider the problem of estimating p in a binomial distribution. Let X be number of successes in a sample of size n. (a) Let the prior distribution of p be given by Beta(3,1), that is T(p) = {3p², 0

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Permutations and Combinations

If there are 5 dishes, they can be relished in any order at a time. In permutation, it should be in a particular order. In combination, the order does not matter. Take 3 letters a, b, and c. The possible ways of pairing any two letters are ab, bc, ac, ba, cb and ca. It is in a particular order. So, this can be called the permutation of a, b, and c. But if the order does not matter then ab is the same as ba. Similarly, bc is the same as cb and ac is the same as ca. Here the list has ab, bc, and ac alone. This can be called the combination of a, b, and c.

Counting Theory

The fundamental counting principle is a rule that is used to count the total number of possible outcomes in a given situation.