# sol4.pdf - MTH5131 Actuarial Statistics Coursework 4 —...

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MTH5131 Actuarial StatisticsCoursework 4 — SolutionsExercise 1.1. LetXbe the number of claims received in a week. To determine the posteriordistribution ofμ, we must calculate the conditional probabilitiesP(μ= 8|X= 7),P(μ=10|X= 7), andP(μ= 12|X= 7),. The first of these isP(μ= 8|X= 7) =P(μ= 8, X= 7)P(X= 7)=P(X= 7|μ= 8)P(μ= 8)P(X= 7)SinceXPoisson(μ),P(X= 7|μ= 8) =e-8877!and since the prior distribution is uniform on the integers 8, 10 and 12:P(μ= 8) =13.SoP(μ= 8|X= 7) =e-8877!×13P(X= 7)=0.04653P(X= 7)Similarly,P(μ= 10|X= 7) =P(X= 7|μ= 10)P(μ= 10)P(X= 7)=e-101077!×13P(X= 7)=0.03003P(X= 7)andP(μ= 12|X= 7) =P(X= 7|μ= 12)P(μ= 12)P(X= 7)=e-121277!×13P(X= 7)=0.01456P(X= 7)Since these conditional probabilities must sum to 1, the denominator must be the sum of thenumerators, soP(X= 7) = 0.04653 + 0.03003 + 0.01456 = 0.09112The posterior probabilities are:P(μ= 8|X= 7) =0.046530.09112= 0.51066P(μ= 10|X= 7) =0.030030.09112= 0.32954P(μ= 12|X= 7) =0.014560.09112= 0.159802. The Bayesian estimate under squared error loss is the mean of the posterior distribution:8×0.51066 + 10×0.32954 + 12×0.15980 = 9.298301
Exercise 2.1. Since the prior distribution ofpis Beta(4,4):f(p)p3(1-p)3Now letXdenote the number of successes from a sample of sizen. ThenXBinomial(n, p).Sincexsuccesses have been observed, the likelihood function is:L(p) =P(X=x) =nxpx(1-p)n-xpx(1-p)n-xCombining the prior PDF with the likelihood function gives:f(p|x)p3(1-p)3×px(1-p)n-x=px+3(1-p)n-x+3Comparing this with the PDF of the Beta distribution, we see that the posterior distributionofpis Beta(x+ 4, n-x+ 4).The Bayesian estimate under all-or-nothing loss is the mode of the posterior distribution, ie thevalue ofpthat maximises the posterior PDF. To find the mode, we need to differentiate the PDF(or equivalently differentiate the log of the PDF) and equate it to zero.Given thatx= 10andn= 25, the posterior ofpis Beta (15,18) and:f(p|x) =Cp14(1-p)17.

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GAMMA, conditional probabilities p