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5. Gamma Distribution. Let U and V be continuous independent random variables and let W = U+ V. Show that the probability density function of W can be written as fw(w) = fv(w- ufo(u) du = fo(w - v)fv(v) du where fo(u) and fv(v) are the density functions of U and V, respectively. Let X1, X2, ..., X, ~ Exp(A) be a sequence of independent and identically distributed ran- dom variables, each following an exponential distribution with common parameter > > 0. Prove that if Y = IM Xi then Y follows a gamma distribution, Y ~ Gamma (n, 1) with the following probability density function: (Av)"-1 fr(y) = de-AY, y 20. (n - 1)! Show also that E(Y) = and Var(Y) =1 A business made a loss during the financial year just ended but has more cash at the end of the year than it did at the beginning. Which of the followingasound be for this? Dividends were lower this year than last year. Some fixed assets were sold during the year. Debtors took longer to pay this year than last. Creditors were lower at the end of this year. [2] 2 A company expects to havemetent liabilities at the financial year end. Raising funds by taking out a short term loan would: A increase the current ratio B reduce the current ratio C have no effect on the current ratio D either increase or decrease the current ratio dapthelingances involved and the extra funds raised [2] 3 Sales revenue should be recognised when goods and services have been supplied; costs are incurred when goods and services have been received. The accounting concept which governs the alberve is accruals concept materiality concept realisation concept dual aspect concept [2] 4 A bond has a value of f1,000 printed on its face. Its current open market value is 1980. Analysts expect the value of the bond to rise to 198bewithinseven days. In your personal opinion it will rise to 1982. What is the nominal value of this bond? A 1980 E982 1985 E1,000 [2]B4 A company is interested in estimating policy lapse rates by age. It conducts an investigation into this, which lasts for the whole of the calendar year 2003. The investigation collects the follodenagfor a sample of policies which are funded by annual premiums: . the age last birthday of the policyholder when the policy was taken out; . the number of premiums the policyholder paid before the policy lapsed. In addition, the number of polifiesaion 1 January each year is available, classified by agdast birthday and yeaalapsed since 1 January 2093,) . (i) State the rate interval in this investigation. ] (ii) Derive an expression for the exposesk interms offy , stating any assumptions you make. [7] (iii) Comment on the reasonableness or otherwise of the assumptions you made in your answer to part (ii). [2] [Total 10]