This practice problem set is to reinforce the topic discussed in this post, the topic of estimating parameters of discrete distributions.
Other posts on parameter estimation focus on continuous distributions – this one and this one. Two practice problem sets, Practice Problem Set 2 and Practice Problem Set 3, are to reinforce these two previous posts.
.
Practice Problem 4A  
The following table gives information on claim frequency data of a group of insureds.
A Poisson distribution with mean is fitted to the claim frequency data.

.
Practice Problem 4B  
The following table gives information on claim frequency data of a group of insureds.
A Poisson distribution with mean is fitted to the claim frequency data using maximum likelihood estimation.

.
Practice Problem 4C 
The probability function for the number of losses for a given insured in a year is given by the following: where and the parameter is unknown. The following shows the numbers of losses for five insureds in one year: 0, 2, 3, 1, 3.

.
Practice Problem 4D  
The observed claim frequency data of a group of policyholders is given in the following table.
A binomial distribution with parameters and is fitted to the given claim frequency data.

.
Practice Problem 4E 
A large group of insureds is made up of two groups – low risk group and high risk group. The annual claim frequency for an insured in the low risk group has a Poisson distribution with mean . The annual claim frequency for an insured in the high risk group has a Poisson distribution with mean . Ten insured are observed for 5 years (5 insureds in each group). Their claim counts are as follows:
High Risk Group: 1, 0, 2, 3, 1 Estimate the parameter using maximum likelihood estimation. 
.
Practice Problem 4F 
The number of claims in a year for an insured has a distribution whose probability function is given by the following: Out of a group of 100 insureds that have been observed for one year, 55 of them have no claims, 25 of them have exactly 1 claim and 20 of them have 2 or more claims.

.
Practice Problem 4G 
Two groups of insureds are pooled for the purpose of maximum likelihood estimation. The number of claims per year for Group 1 follows a binomial distribution with parameters and . The number of claims per year for Group 2 follows a binomial distribution with parameters and . In observing these two groups for 3 years, there are 15 claims from Group 1 and 28 claims from Group 2. Estimate the parameter using maximum likelihood estimation. 
.
Practice Problem 4H 
The number of claims from 5 policyholders are:
A zerotruncated geometric distribution is fitted to the claim data using maximum likelihood estimation. Determine the estimated probability that the number of claims is at least 2. 
.
Practice Problem 4I  
The observed claim frequency data for a group of 105 insureds is given below.
Which of the (a,b,0) distributions (Poisson, binomial, negative binomial) is the most appropriate fit to the claim frequency data? Answer this question from the following two angles.

.
Practice Problem 4J 
When fitting a binomial distribution with both parameters unknown, the maximum likelihood estimation using loglikelihood profile is demonstrated in Example 7 and Example 8 in this post. Show that this approach does not work for the claim frequency data in Problem 4D. 
Problem  Answer 

4A 

4B 

4C 

4D 

4E 

4F 

4G 

4H 

actuarial practice problems
Dan Ma actuarial
Daniel Ma actuarial
Daniel Ma Math
Daniel Ma Mathematics
Actuarial exam
2019 – Dan Ma