Download Math 395 Spring 2005 Quiz 2 March 18, 2005 Due Back March 25

Math 395
Spring 2005
Quiz 2
March 18, 2005
Due Back March 25, 2005 Name:______________________
You may refer to any books or notes that you wish and you may work with other. Each student is
responsible for his or her own solutions, however, so don’t blindly accept any other source without
satisfying yourself that it’s correct. Show all of your supporting work, on separate pages if necessary.
1.
S is a compound Poisson frequency distribution with λ=10 and a secondary distribution that has
non-zero probability only at the values 2, 3, 4, 5. E[S]= 30, Var[S]=100, and the coefficient of
kurtosis of S is 3.1744. Find the expected value for the frequency of events S having each of the
secondary distribution values 2, 3, 4, and 5.
Expected # of 2’s____3’s_____4’s______5’s_______
2.
You have assumed that N is a Poisson variable with parameter λ representing the number of paid
claims in year from a certain insurance coverage. Assume that there will be a 60% increase in
exposure. If you want to introduce a deductible so as to bring the expected number of paid claims
back down to what it was before the increase in exposure, what percentage of the total number of
claims must you eliminate with the deductible?
Eliminate_________% of claims
Unfortunately, it turns out that the original N was negative binomial with parameters βr=λ, instead
of Poisson. With the same exposure and deductible changes described above, what is the resulting
distribution (including its parameters)?
Distribution__________________
Name:______________________________
3.
Let S be a compound Poisson-Poisson frequency distribution with primary frequency λ 1 = .05 and
secondary frequency λ 2 =5. What is the mode of S? (the most probable single number of events.)
Mode_______
4.
If the parameters are the same in both cases, under what conditions will a compound negative
binomial-binomial distribution have a larger variance than a compound binomial-negative
binomial distribution?
Conditions:____________________
____________________
____________________
Name:______________________________
5.
If two otherwise identical groups of insurance risks whose numbers of claims each follow negative
binomial distributions have the relationship that the coefficient of skewness of A is half the
coefficient of skewness of B, what is the size of group A in relation to group B?
A is ___________times larger/smaller than B
6.
Let the number of claims for some accident insurance coverage follow a Poisson distribution with
λ=45,000. The marketing department notices that in one out of every 15 claims the accident
victim uses the services of an acupuncturist, which is not covered so the claim department denies
payment. The marketing department decides to offer a coverage supplement that will pay for the
services of an acupuncturist in the event of accident. What is the theoretical distribution
(including parameters) for the number of acupuncture claims?
Distribution_____________________________
As the actuary would you go ahead and set prices for the acupuncture coverage supplement based
on this theoretical distribution? Discuss.
Name:___________________________
7.
The random variable N is in the (a,b,0) class. Let 1/x be N’s coefficient of variation. Let y be the
square root of (N’s variance, plus the ratio of N’s variance to its mean, minus 1). Assume that
(x+y)2=x2 + y2. Finally, assume that the probability that N=4 is 1/32.
a.
What is the conditional probability that N=5,452, conditional on N > 5,447?
Conditional Probability=___________
b.
Tell me everything that can you say about the conditional probability that N=5,449,
conditional on N> 5,447?