please help

Sampling from a sum of compound Poisson distributions Recall Theorem 2.6: Suppose S, has a compound Poisson distribution with parameters > > 0 and fx. ( . ), for i = 1, ..., n, and define S = S, + ... + 5.. If S, and S, are independent, for all i # j, then S has a compound Poisson distribution with parameters: A - d and fv(.) = E- fx(.). 1= 1 Question 1 Write down an expression for the random variable Y in terms of its constituent parts, and describe, in general, how to generate a single observation of this random variable. [3] Question 2 Given that n = 2, with X1 ~ Gamma(a = 2, A = 0.001) and X2 ~ Pareto(a = 2.5, A = 1500), write R code for generating a single observation of the random variable Y. [3] Hint: use either the if function, which has the following basic structure: > if (user -defined test condition) { user-defined output if test condition is true } else { user-defined output if test condition is false or the ifelse function, which has the following basic stucture: > ifelse (user-defined test condition , user -defined output that the ifelse function is set equal to if test condition is true, user -defined output . . . if test condition is false) Question 3 Given that A1 = 25 and Ag = 75, modify your code from Question 2 so that it can be used within a for loop to generate a single observation of the random variable S. Please run the code set. seed (123) before randomly generating your variables (the set . seed function tells R to use a particular root for generating random variables, the result of which is that your output will be repeatable-and verifiable by me as long as the same seed is applied). [4] Question 4 (Bonus) By generating a sample of size /V = 10000 of the random variable S, estimate the amount of reserves an insurance company should hold to ensure that the probability of being unable to cover their losses is 0.5%. Please be sure to run the code set . seed (123) again before randomly generating your sample. [4]