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Suppose that the random variable X follows an exponential distribution with probability density function f(x=20-", ocx0). Define a new random variable V = X 3 (i) (a) Show that the cumulative density function of V is given by Fy()= 1-exp(-2y ) y20 and hence, or otherwise, find the probability density function of Y. (b) Explain how you would simulate a value of Y' given a value & from the uniform ((0, 1) distribution. [7] (ii) (a) Find an expression for the maximum likelihood estimator of the parameter A, using a sample >y. V2. .... >.. from the distribution of Y. (b) Eight observed values of the random variable Y' are given below: 0.72 1.15 1.26 1.03 1.69 1.30 1.42 1.15 Calculate the maximum likelihood estimate of 2 using these values. [6] (iii) (a) The hazard function of a continuous random variable T is defined as horj = J () where f(@) denotes the probability density function and S() S(r) denotes the survival function defined as S (1) = P(T > (). Derive the hazard functions of the random variables X and Y defined above. (b) If a random variable T represents the lifetime of an individual, then the hazard function hit). as defined in part (iii)(a), gives the instantaneous mortality rate (that is, the force of mortality) at time / for that individual State (with reasons) which of the two random variables (X and Y) you would use to model the lifetime of pensioners for a period of time longer than one year, basing your answer on the form of the corresponding hazard functions derived in part (iii)(a). [5] [Total 18]On a small distant planet lives a race of aliens. The aliens can die in one of two ways, either through illness, or by being sacrificed according to the ancient custom of the planet. Aliens who die from either cause may, some time later, become zombies. (i) Draw a multiple-state diagram with four states illustrating the process by which aliens die and become zombies. labelling the four states and the possible transitions between them. [2] (ii) Write down the likelihood of the process in terms of the transition intensities, the numbers of events observed and the waiting times in the relevant states, clearly defining all the terms you use. [4] (lii) Derive the maximum likelihood estimator of the death rate from illness. [3] The aliens take censuses of their population every ten years (where the year is an "alien year", which is the length of time their planet takes to orbit their sun). On I January in alien year 46,567, there were 3,189 live aliens in the population. On I January in alien year 46,$77 there were 2,811 live aliens in the population. During the intervening ten alien years, a total of 3.690 aliens died from illness and 2.310 were sacrificed, and the annual death rates from illness and sacrifice were constant and the same for each alien. (iv) Estimate the annual death rates from illness and from sacrifice over the ten alien years between alien years 46,567 and 46,577. [2] The rale at which aliens who have died from either cause become zombies is 0.1 per alien year. (v) Calculate the probabilities that an alien alive in alien year 46,567 will, ten alien years later: (a) still be alive (b ) be dead but not a zombic (7] [Total 18]