m A single ob5e rvation, .v, is drawn from a distribution with the probability density function: til1 tics-dd flxldl= otherwise The prior PDF of i9 is given by: fidi = denialHi, ii a D Derive an expression in terms of it for the Eiayesian estimate of i? under absolute error loss. [4] Ti A proportion p of packets of a rather dull breakfast cereal contain an exciting toy {independently I from paclzet to packet}. An actuary has been persuaded by his children to begin buying packets of this cereal. His prior beliefs about p before opening any packets are given by a uniform distribution on the interval [L11]. It turns out the first toy is found in the o1 th packet of cereal. iii Specify the posterior distribution of p after the first toy is found. [3] A further toy was found after opening another a; packets, another toy after opening another :13 packets and so on until the fth toy was found after opening a grand total of :11 + n2 + n3 + an + n5 packets. iii} Specify the posterior distribution of p after the fifth toy is found. [2] {iii} Show the Bayes' estimate of p under quadratic loss Is not the same as the maximum likelihood estimate and comment on this result. [5] [Total 1D] 3 An actuary has a tendency to be late for work. If he gets up late then he arrives at work I I minutes late where I is exponentially distributed with mean 15. If he gets up on time then he arrives at worl: if minutes late where 'i" is uniformly distributed on [1125]. The ofce manager believes that the actuary gets up late one third of the time. Ealculate the posterior probability that the actuary did in fact get up late given that he arrives more than 2i] minutes late at work. [5] The next three questions are based on the following figures from ABC Ltd's accounts. Unless stated otherwise the figures are as at the end of the financial year: E 000 Sales 200 Inventories at the beginning of the year 30 Inventories at the end of the year 20 Purchases 130 Administrative expenses 15 Trade receivables 40 Prepayments 3 Cash 17 Bank overdraft 12 Trade payables 10 Accruals 2 ABC Ltd has issued 200,000 50p ordinary shares. During the year, it paid a dividend of 4.5p. 1 The average inventory turnover period of ABC Ltd in days is: 36 days B 46 days 65 days 78 days [2] 2 ABC Ltd's current ratio is: 2.5 : 1 3.3 : 1 3.5 : 1 4.0 : 1 [2] 3 ABC Ltd's dividend cover (ignoring taxation) is: 2.5 times 3.1 times 5.0 times 6.7 times [2]