Solve these challenging questions. Show all the steps and explanations

(i) Use the following uniform(0,1) random numbers 0.9236 , 0.2578 and a suitable table of probabilities to simulate two observations of the random variable X. where X ~ N(200.100). 131 (ii) Use the following uniform(0.1) random numbers 0.3287 . 0.9142 to simulate two observations of the random variable Y, where I has an exponential distribution with mean 100. [3] [Total 6] A random sample of four insurance policies of a certain type was examined for each of three insurance companies and the sums insured were recorded. An analysis of variance was then conducted to test the hypothesis that there are no differences in the means of the sums insured under such policies by the three companies. The total sum of squares was found to be SS = 420.05 and the between-companies sum of squares was found to be SS; = 337.32. (i) Perform the analysis of variance to test the above hypothesis and state your conclusion. (ii) State clearly any assumptions that you made in performing the analysis in (i). [2] (iii) The plot of the residuals of this analysis of variance against the associated fitted values, is given below. Residuals 18 20 22 24 26 28 30 32 Fitted Comment briefly on the validity of the test performed in (i), basing your answer on the above plot. [2]Define a Poisson process [2] A bus route in a large town has one bus scheduled every 15 minutes. Traffic conditions in the town are such that the arrival times of buses at a particular bus stop may be assumed to follow a Poisson process. Mr Bean arrives at the bus stop at 12 midday to find no bus at the stop. He intends to get on the first bus to arrive. (ti] Determine the probability that the first bus will not have arrived by 1.00 pm the same day. [2] The first bus arrived at 1.10 pm but was full, so Mr Been was unable to board it. (iii) Explain how much longer Mr Bean can expect to wait for the second bus to arrive. [1] (iv) Calculate the probability that at least two more buses will arrive between 1.10 pm and 1.20 pm. 121 [Total 7]