Requirements. Address all questions appropriately. Only correct answers please.

Waiting times in a post office queue have an Exp(1) 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 find the maximum likelihood estimate of 1.Claims (in f000s) on a particular policy have a distribution with PDF given by: f (x) = 2cre-er Seven of the last ten claims are given below: 1.05, 3.38, 3.26, 3.22, 2.71, 2.37, 1.85 The three remaining claims were known to be greater than 16,000 (and were passed onto a reinsurer). Find the maximum likelihood estimate of c.Independent random samples of size n, and n, are taken from the normal populations N(1,67) and N(#2,57) respectively. (i) Write down the sampling distributions of X, and X, and hence determine the sampling distribution of X1 - X2, the difference between the sample means. (ii) Now assume that of = o} =02. (a) Express the sampling distribution of X] - X, in standard normal form. (b) State the sampling distribution of (m - 1)SF + (12 - 1)$2 (c) Using the A(0,1) distribution from (a) and the y distribution from (b), apply the definition of the f distribution to find the sampling distribution of X - X, when o' is unknown.1. Consider the following constrained two-period optimization problem: max *(c1, c2) = - exp(-yci) - Bexp(-702) (1) subject to: a +- R (2) where ~ >0 is the coefficient of absolute risk aversion (CARA), B e (0, 1) is the discount factor, (c1, c2) is the optimal bundle of consumption in periods 1 and 2, I is the given total income during the two periods, R is the gross interest rate. (a) Set up the Lagrangian function for the above optimization problem, and derive the first order conditions