PPlease address the following questions keenly. Thank you
Let the random variables (X, Y) have the joint probability density function fx,y(x, y) = exp(-(x+y)}, x>0.y>0. Derive the marginal probability density functions of X and Y', and hence determine ( giving reasons) whether or not the two variables are independent. (3] (ii) Derive the joint cumulative distribution function Fr r(x, y). [2] [Total 5] Let X be a random variable with probability density function given by f(r) =>AZ, OcrcA. Find an unbiased estimator of 0, based on a single observation of X. [4] A random sample of size n is taken from an exponential distribution with parameter 2., that is, with probability density function Determine the maximum likelihood estimator (MLE) of 2.. (3] Claim sizes for certain policies are modelled using an exponential distribution with parameter 2. A random sample of such claims results in the value of the MLE of ) as A = 0.00124. A large claim is defined as one greater than (4,000 and the claims manager is particularly interested in p, the probability that a claim is a large claim. (ii) Determine p , the MLE of p, explaining why it is the MLE. (3] [Total 6]A scientific investigation involves a linear regression with the usual assumptions that the response variable y follows a normal distribution with mean a + Box and variance G3. Twenty data points were recorded, corresponding to four observations of y at x = 1, three observations of yal x = 2, six observations of y at x = 3, and seven observations of y atx =4. The resulting means of these sets of y observations are given in the table below. WN no. of y's mean of y's 18.6 21.7 23.2 27.1 (i) Determine the fitted regression line of y on r. (5] (ii) Suppose that you have been asked to provide a 95% confidence interval for the slope coefficient. (a) Comment briefly on any problems you might encounter in the computation of the required confidence interval (b) Indicate briefly any further information that you would need in order to overcome these problems. (3] [Total 8] The table below shows a bivariate probability distribution for two discrete random variables X and Y: X-0 X-1 X-2 Y = 1 0.15 0.20 0.25 Y = 2 0.05 0.15 0.20 Find the value of E[X Y = 2]. [3] In a group of motor insurance policies issued by a company, 80% of claims are made on comprehensive policies and 20% are made on third-party-only policies (1) Calculate the average amount paid out on a claim, given that the average amount paid out by the company on a comprehensive policy claim is $1,650, and the average amount paid out on a third-party-only policy claim is 8625. [1] (ii) Calculate the total expected amount paid out in claims by the company in one year, given that the total number of policies is 150,000 and, on average, the claim rate is 0.15 claims per policy per year. (21 [Total 3]
