1. The numbers of claims which have arisen in the last twelve years on 60 motor car policies (continuously in force over this period) are shown (sorted) below 0 0 00 0 0 0 0 0 0 0 0 1 1 1 11 1 11...

please send solution of Q1 only handwritten solution accepted


Extracted text: 1. The numbers of claims which have arisen in the last twelve years on 60 motor car policies (continuously in force over this period) are shown (sorted) below 0 0 00 0 0 0 0 0 0 0 0 1 1 1 11 1 11 111111 22 2 2 2 2 2 2 2 22 3 3 3 33 3 3 3 3 3 44 44 4555 5 6 6 6 7 (a) Draw a barplot of the number of claims. (b) Calculate the median, mode and mean of the data. (c) Calculate the interquartile range and the standard deviation of the data. 2. Let X1, X2, ,X, be a random sample of size n from an Exp(1/0) distribution (i.e with mean 0 and variance 62). (a) Derive an expression for E(X) and Var(X) in terms of 0 where X is the mean of the random sample. [3] (b) Express MSE(X) in terms of Var(X) and Bias (X) and hence find an expression for MSE(X) in terms of 0. (c) Consider the family of estimators for 0 of the form Y aX. Find MSE(aX) and determine the value of a' such that Y = a'X is the most efficient estimator of the form Y = aX. 3. Let X1, X2, ,X, be a random sample from the probability distribution 0;0<><1 f(z,0)="otherwise" where="" the="" parameter="" 0=""> 0. (a) Find E [X] for this distribution (b) Use the method of moments (MME) to find an estimator for 0 (c) Find the maximum likelihood estimator (MLE)for 0. (d) Calculate the estimate using the MME and MLE estimators when = 0.10, r = 0.22, r3 0.54, = 0.36