4 Questions, use excel and diagrams where necessary. Thankyou
Q2 Let X1, X2, ...;Xn be a random sample consisting of independent random variables with mean u and variance oz. Consider the sample mean: X = n In your answer you may denote u by mu, o' by sigma2 and X by Xbar. (i) Write down the expected value of X. [1] (ii) Derive the variance of X. [2] (iii) Comment on the variance of variable X as compared to the variance of X. [1] An actuary is interested in exploring the difference in the size of claim losses from two insurance portfolios, and can take samples of claims from these portfolios. (iv) Explain how the answer to part (iii) can affect the precision of the actuary's comparison. [2] [Total 6]Q1 A survey showed that 40% of investors invest in at least two companies in order to diversify their risk. Let X be the random variable denoting the number of investors who have invested in more than one company in a random sample of 300 investors. (i) Determine the distribution of X. [2] (ii) Calculate an approximate probability that X is greater than 100. [2] [Total 4]Q3 X and Y are discrete random variables with joint distribution as follows: X=0 X= 1 X = 3 Y=-1 0.08 0.03 0.00 Y=0 0.03 0.12 0.20 Y =3 0.11 0.11 0.06 Y =4.5 0.04 0.20 0.02 (i) (a) Identify which one of the following options gives the correct value of the expectation E(Y [ X = 1) : [2] (Al) 1.3789 (A2) 2.6087 (A3) 3.1398 (A4) 4.0945 (b) Identify which one of the following options gives the correct value of the variance var(X | Y = 3) : [2] (Al) 1.2487 (A2) 0.9832 (A3) 1.9388 (A4) 2.2235 (ii) Calculate the probability functions of the marginal distributions for X and Y. [2] (iii) Determine whether X and Y are independent. [2] [Total 8]Q4 An actuary is asked to check a linear regression calculation performed by a trainee. The trainee reports a least squares slope parameter estimate of b = 13.7 and a sample correlation coefficient r = = 0.89. (i) Justify why this suggests that the trainee has made an error. [2] In a different simple linear regression model, a histogram of the residuals is shown below. 100 150 200 Frequency 50 -0.4 -0.2 0.0 0.2 0.6 0.8 Residuals (ii) Comment on the validity of the assumptions of the linear model. [2] The following pairs of data are available: 2 8 -1.35 - 4.96 - 9,20 -13.1. -16.70 -21.23 - 25.14 28.44 - 33.68 -37.39 for which y = -19.124, (VI - 9)2 = 1,329.523, M (x1 - 8)2 = 82.5 [(x - 2)(1-7) = -331.05 A linear model of the form y = a + bx + e is fitted to the data, where the error terms (e) independently follow a N(0, a") distribution, and where a, b and o' are unknown parameters. (iii) Determine the fitted line of the regression model. [3] (iv) (a) Identify which one of the following options gives the correct estimate of the variance o of the model. [1] (A1) 0.612 (A2) 1.098 (A3) 0.971 (A4) 0.139 (b) Identify which one of the following options gives the correct estimate of the variance of the predicted mean response if x = 11. [2] (Al) 0.161 (A2) 0.085 (A3) 0.287 (A4) 0.309 (c) Calculate a 95% confidence interval for the predicted mean response if x = 11. [2] (v) Comment on the width of a 95% confidence interval for the predicted mean response if x = 3.5, as compared to the width of the interval in part (iv), without calculating the new interval. [2] [Total 14]
