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1 A random sample from a Poisson(//) distribution is as follows: 4, 2, 7, 3, 1, 2, 5, 4, 0, 2 Calculate the method of moments estimate for /. 2 Using the method of moments, estimate the mean and variance of the heights of 10-year old children, assuming these conform to a normal distribution, based on a random sample of 5 such children whose heights are: 124cm, 122cm, 130cm, 125cm and 132cm. 3 Waiting times in a post office queue have an Exp(/) distribution. Ten people had waiting times (in minutes) of: 1.6 0.9 1.1 2.1 0.7 1.5 2.3 1.7 3.0 3.4 A further six people had waiting times of more than 4 minutes. Based on these data calculate the maximum likelihood estimate of 1. 4 The number of claims arising in a year on a certain type of insurance policy has a Poisson style distribution with parameter 2. The insurer's claim file shows that claims were made on 238 policies during the last year with the following frequency distribution for the number of claims: Number of claims Frequency 174 50 W 10 4 4 2 5 0 No information is available from the policy file, that is, only data concerning those policies on which claims were made can be used in the estimation of the claim rate 1. (This is why there is no entry for the number of claims being 0 in the table.) (i) Show that the truncated probability function is given by: P(X = x)=_ x =1,2,3,... [3] x!(1-e-2) (ii) Hence show that both the method of moments estimate and the MLE of / satisfy the equation 1=x(1-e *), where x is the mean number of claims for policies that have at least one claim. [7](iii) Solve this equation, by any means, for the given data and calculate the resulting estimate of 1 to two decimal places. [3] (iv) Hence, estimate the percentage of all policies with no claims during the year. [1] [Total 14] 5 Determine the mean square error of / = X which is used to estimate the mean of a N(#,of ) distribution based on a random sample of n observations. 6 Suppose that unbiased estimators X, and X2 of a parameter & have been determined by two style independent methods, and suppose that var(X])=o and that var(X2)=do , where >0. Let Y be the combination given by Y =a X1 + /X2, where a and / denote non-negative weights. (i) Derive the relationship satisfied by a and / so that Y is also an unbiased estimator of 0. [2] (ii) Determine the variance of Y in terms of $ and o if, additionally, the weights are chosen such that the variance of Y is a minimum. [4] 7 A random sample X1, *2,...*, is taken from a population, which has the probability distribution style function F(x) and the density function f(x). The values in the sample are arranged in order and the minimum and maximum values XM/w and XMAX are recorded. (i) Show that the distribution function of XMAX is [F(x)]", and find a corresponding formula for the distribution function of XMIN . [3] The original distribution is now believed to be a Pareto(a, 1) distribution, ie the probability density function is: f(x) =- (1+x18+1' *20 (Wi) Find the distribution function of X , and hence find the distribution function of XMAX . [2] (iti) Show that the probability density function for the distribution of XMIN , is: fx MIN (x) = [2] (1 + x)+1 X20 (iv) A random sample of 25 values gives a sample value for XM of 23. Use the distribution of XMIN to obtain a maximum likelihood estimate of a. [3] (v) The same random sample gives a value of XMAX of 770. Obtain an equation for the maximum likelihood estimator of a using XMAX - Comment on the difficulty of solving this equation. [3]
