An individual picked at random from a population has a propensity to have accidents that is modelled by a random variable Y having the gamma distribution with shape parameter α and rate parameter β. Given Y = y, the number of accidents that the individual suffers in years 1, 2, . . . , n are independent random variables X1, X2, . . . Xn each having the Poisson distribution with parameter y. (a) Write down a function f so that the joint distribution of Y, X1, . . . , Xn can be described via P(a ≤ Y ≤ b, X1 = k1, X2 = k2 . . . Xn = kn) = Z b a f(y, k1, k2, . . . kn)dy and derive from this expression that, for your choice of f, Y has the Gamma distribution, and that conditionally on Y = y, X1, X2, . . . Xn are independent, each having the Poisson distribution with parameter y. (b) Find the conditional distribution of Y given that X1 = k1, X2 = k2, . . . , kn. (c) An insurance company has observed the number of accidents that an individual has suffered on each of n years and wishes to predict the number of accidents that individual will experience in the next year. To this end let Xn+1 be a further random variable so that, conditionally on Y = y, X1, X2, . . . Xn, Xn+1 are independent, each having the Poisson distribution with parameter y. Write down the value of the conditional expectation E[Xn+1|X1, X2, . . . Xn, Y ], and hence determine E[Xn+1|X1, X2, . . . Xn].
4.4. An individual picked at random from a population has a propensity to have accidents that is modelled by a random variable Y having the gamma distribution with shape parameter α and rate parameter β. Given Y = y, the number of accidents that the individual suffers in years 1, 2, . . . , n are independent random variables X1, X2, . . . Xn each having the Poisson distribution with parameter y.
(a) Write down a
(b) Find the conditional distribution of Y given that X1 = k1, X2 = k2, . . . , kn.
(c) An insurance company has observed the number of accidents that an individual has suffered on each of n years and wishes to predict the number of accidents that individual will experience in the next year. To this end let Xn+1 be a further random variable so that, conditionally on Y = y, X1, X2, . . . Xn, Xn+1 are independent, each having the Poisson distribution with parameter y. Write down the value of the conditional expectation E[Xn+1|X1, X2, . . . Xn, Y ], and hence determine E[Xn+1|X1, X2, . . . Xn].
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