QUESTION 3 TOTAL 20 marks Let n 2 2 be a positive integer. Let X1, ..., X, be i.i.d. exponential random variables with parameters 1 each. 1. n books are placed in a random order on a bookshelf. (a) Find the probability p(n) that two given books are placed side by side. Justify your answer. Your answer should depend on n only. 3 marks (b) Set n = 2 in your formula for p(n). If you obtain p(2) = 1 then you get . . . . . 1 mark 2. n books of weights X1,..., Xn are placed in a random order on a bookshelf such that the order is independent of the weights. (a) Find the probability that the two lightest books are placed side by side. 3 marks (b) Let S,, be the sum of all the weights and let g, be its probability density function. Explain why 8n ( x ) = e- (-ygn-1(y)dy, x20, () and verify that 8n (x) = ; 1 (n - mukh -lex satisfies (J). 3 marks 3. Recall that [(a) := 2lexdx, a>0. Using this, compute the integral where 1 > 0. 3 marks 4. Define, for each a > 0, the function fra(10):= No 14-ledx, x20. Using your result for the integral in Item 3, explain why fa,a is a probability density function on [0, co). 3 marks 5. Using Items 4 and 5 derive a formula for the moment generating function of a random variable X with probability density function fix- 4 marks