Question 37 may 2001

[Gore Tree Lover]

Can anyone explain how to do this problem? These posterior distributionns confuse me? Seems like it should be easy but I just can't get it.

[New at pd]

What is the question? I don't have the exams in front of me.

[Dr T Non-Fan]

I think more people are confused about which course's exam you're discussing. If only...
(Yes, I'm hollow-victory-dancing.)

[Gore Tree Lover]

Sorry, Exam 4 (spring 2001). If only.

The question asked to calculate the probability of a person earning more than 50,000 if they had no claims in the previous year.

I don't have the exam in front of me right now either.

[Dr T Non-Fan]

If you're online, then you could at least have it minimized, from the SOA site.

Posterior probability question.
Number of claims for an employee follows Poisson with mean (100-p)/100;
p = salary in thousands.
Distribution of p is uniform along (0,100].

How to solve: I don't know. Already have credit for Exam 4. Don't have a text in front of me.

[New at pd]

This is a straight-forward application of Bayesian credibility:

First, not that for prior distributions that are uniform distributions are negligible when calculating the posterior distribution.

A Poisson dist of mean lambda has
Pr(N=0 | lambda) = exp(-lambda) =
exp(-(1 - p/100))

The post. of p is then exp(-(1 - p/100)), divided by the integral of the above over p.

Then, just integrate the post. dist from 50K to 100K to get the probability that the employees salary is over 50K.

[Gore Tree Lover]

The Soa solution talked about a normalizing Constant. What is that? I have been doing calculus for many many years and do not recall that phrase or maybe I just have a bad memory.

[New at pd]

The normalizing constant that they are talking about is the integral of

Pr(p | no claims) * Pr(p), over all p

This integral is usually not 1, and is used to divide Pr(p | no claims) * Pr(p) so that the cumulative probablity of the posterior distribution is 1.