Solve appropriately. It's an assignment and am stuck. Please help

(i) A random variable I has a Poisson distribution with parameter O but there is a restriction that zero counts cannot occur. The distribution of Y' in this case is referred to as the zero-truncated Poisson distribution. (a) Show that the probability function of Y is given by P()) - - v!(1-e-0) ( - 1,2, ...). (b) Show that E[Y]- 0/(1-2-") [4] (ii) (a) Let y1-.... y, denote a random sample from the zero-truncated Poisson distribution. Show that the maximum likelihood estimate of 9 may be determined by the solution to the following equation: 1-20 0 and deduce that the maximum likelihood estimate is the same as the method of moments estimate. (b) Obtain an expression for the Cramer-Rao lower bound (CRIb) for the variance of an unbiased estimator of 0. 19] (iii) The following table gives the numbers of occupants in 2.423 cars observed on a road junction during a certain time period on a weekday morning. Number of occupants 2 3 4 5 6 Frequency of cars 1,486 694 195 37 10 The above data were modelled by a zero-truncated Poisson distribution as given in (1). The maximum likelihood estimate of 0 is 0 = 0.8925 and the Cramer-Rac lower bound on variance at 6 = 0.8925 is 5.71 1574 x 10 *(you do not need to verify these results.) (a) Obtain the expected frequencies for the fitted model, and use a y= goodness-of-fit test to show that the model is appropriate for the data. (b) Calculate an approximate 95% confidence interval for ( and hence calculate a 95% confidence interval for the mean of the zero-truncated Poisson distribution. [9] [Total 22]An insurance portfolio contains policies for three categories of policyholder: A, B and C. The number of claims in a year, N, on an individual policy follows a Poisson distribution with mean A. Individual claim sizes are assumed to be exponentially distributed with mean 4 and are independent from claim to claim. The distribution of 2., depending on the category of the policyholder, is Category Value of A Proportion of policyholders 20% 60% 20% Denote by S the total amount claimed by a policyholder in one year. Prove that E(S) = E[E(SA)]. [2] (ii) Show that E(5 ).) = 42. and Var($ 2)=327 . [2] (Hii) Calculate F(S). [2] (iv) Calculate Var( S). 12] [Total 8] The following information is available for a motor insurance portfolio: The number of claims settled: Development Year Accident year ! 2006 442 151 50 2007 623 111 2008 681 The cost of settled claims during each year (in 000's): Development Year Accident year 2 2006 6321 1901 701 2007 7012 2237 2008 7278 Claims are fully run off after year 2. Calculate the outstanding claims reserve using the average cost per claim method with grossing up factors. Inflation can be ignored. [ot]