3. (a) Consider the classical risk model, where the surplus of an insurance company can be modelled as N() U(t) = 1+ c -Lx, U(0) =u > 0, i=1 where N(t) is a Poisson process, with parameter 1, representing the number of the claims and {X;}i>1 are i.i.d. random variables, independent of N(t). The density of the claim amounts is given by 21 7 fx(x) = + 2e-5, > 0. 5 Also, you are given that the intensity of the Poisson process is 1 = 4 and the premium received per unit time is c=3. (i) Write an R programme to find the relative security loading for this model. [4 marks (ii) Write an R programme to plot the ruin probability. [6 marks] (b) Consider the classical risk model, in which the density of the claim amounts is given by fx(1) = 2.ce r> 0. Also, you are given that the intensity of the Poisson process is 1 = 2 and the premium received per unit time is c=6. (i) Find the moment generating function of X. [3 marks) (ii) Write an R programme to find the adjustment coefficient for this model. [7 marks] 3. (a) Consider the classical risk model, where the surplus of an insurance company can be modelled as N() U(t) = 1+ c -Lx, U(0) =u > 0, i=1 where N(t) is a Poisson process, with parameter 1, representing the number of the claims and {X;}i>1 are i.i.d. random variables, independent of N(t). The density of the claim amounts is given by 21 7 fx(x) = + 2e-5, > 0. 5 Also, you are given that the intensity of the Poisson process is 1 = 4 and the premium received per unit time is c=3. (i) Write an R programme to find the relative security loading for this model. [4 marks (ii) Write an R programme to plot the ruin probability. [6 marks] (b) Consider the classical risk model, in which the density of the claim amounts is given by fx(1) = 2.ce r> 0. Also, you are given that the intensity of the Poisson process is 1 = 2 and the premium received per unit time is c=6. (i) Find the moment generating function of X. [3 marks) (ii) Write an R programme to find the adjustment coefficient for this model. [7 marks]