Assorted Exam-Style Questions on Compound Poisson Processes, Normal Approximations, Force of Mortality, and Multiple Forces of Decrement

This section of sample problems and solutions is a part of The Actuary's Free Study Guide for Exam 3L, authored by Mr. Stolyarov. This is Section 62 of the Study Guide. See an index of all sections by following the link in this paragraph.

When determining the variance of the sum of independent Poisson random variables, you need to simply add the variances of each Poisson random variable. Note that the Poisson random variable X has the same variance as the Poisson random variable -X.

When doing some of the problems in this section, recall that the Poisson distribution is a discrete distribution and so when it is approximated by the continuous normal distribution, a continuity correction is required.

From Section 59, when you have an integer g, then, by the continuity correction,

Pr(More than g) = Pr(At least g + 1) = Pr(g + 1 or more) = 1 - Φ((g + 0.5 - μ)/σ).

When you have an integer g, then, by the continuity correction,

Pr(Less than g) = Pr(At most g - 1) = Pr(g - 1 or less) = Φ((g - 0.5 - μ)/σ).

A compound Poisson process S(t) occurs when some Poisson process N(t) accompanies some random variable X. For instance, N(t) can describe the frequency of losses of a particular kind, and X can describe the severity of each loss.

The expectation and variance of S(t) can be calculated as follows.

E(S(t)) = E(X)*E(N(t)) and Var(S(t)) = E(N(t))*Var(X) + Var(N(t))*E(X) ² = E(N(t))*E(X²).

A useful expression relating survival probabilities to the force of mortality is

_np_x = exp(-_x^x+n∫µ(t)*dt). In many situations, this formula gives an easy way of calculating _np_x given an expression for µ(t).

Sources: Broverman, Sam. Actuarial Exam Solutions - CAS Exam 3L - Spring 2008.

Daniel, James W. 2008. Poisson Processes (and Mixture Distributions).

Original Problems and Solutions from The Actuary's Free Study Guide

Problem S3L62-1. Similar to Question 11 from the Casualty Actuarial Society's Spring 2008 Exam 3L. This question is also an excellent review of the concepts in Section 50.