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2019年4月英国精算师-精算统计真题 4 April 2019 (am)

Subject CS1B – Actuarial Statistics Core Principles

1 The following data represent the average total number of marks obtained for a particular exam, observed over seven exam sessions that had been administered by a professional examination body:

87 53 72 90 78 85 83

(i) Enter these data into R and compute their sample mean and variance.

(ii) Investigate whether the Poisson model is appropriate for these data, by calculating the sample mean and sample variance of 10 Poisson samples having the same size and mean as the sample given above.

2 Consider the n = 30 independent and identically distributed observations ( y1, y2, …, yn) given below from a random variable Y with probability distribution (i) (a) Plot the posterior probability density function of θ for values of θ in the interval [3.2, 6.8] and assuming a = 0.01.

[Hint: the range of values of θ can be obtained in R by seq(3.2, 6.8, by = 0.01).]

(b) Carry out a simulation of N = 5,000 posterior samples for the parameter θ.

(ii) Plot the histogram of the posterior distribution of θ.

(iii) Calculate the mean, median and standard deviation of the posterior distribution of θ.

Two possible values for the true value of parameter θ are θ =15 and θ = 5.

(iv) Comment on these two values based on the posterior distribution of θ plotted in part (ii) and summarised in part (iii). 