Practice Problems
Practice Problem Set 12 – (a,b,1) class
The practice problems in this post are to reinforce the concepts of (a,b,0) class and (a,b,1) class discussed this blog post and this blog post in a companion blog. These two posts have a great deal of technical details, especially the one on (a,b,1) class. The exposition in this blog post should be more accessible.
Notation: for whenever is the counting distribution that is one of the (a,b,0) distributions – Poisson, binomial and negative binomial distribution. The notation is the probability that a zerotruncated distribution taking on the value . Likewise is the probability that a zeromodified distribution taking on the value .
Practice Problem 12A 

Practice Problem 12B 
This problem is a continuation of Problem 12A. The following is the probability generating function (pgf) of the Poisson distribution in Problem 12A.

Practice Problem 12C 
Consider a negative binomial distribution with and .

Practice Problem 12D 
The following is the probability generating function (pgf) of the negative binomial distribution in Problem 12C.

Practice Problem 12E 
This is a continuation of Problem 12C and Problem 12D.

Practice Problem 12F 
This problem is similar to Problem 12E.

Practice Problem 12G 
Suppose that the following three probabilities are from a zerotruncated (a,b,0) distribution.

Practice Problem 12H 
Consider a zeromodified distribution. The following three probabilities are from this zeromodified distribution.

Practice Problem 12I 
For a distribution from the (a,b,0) class, you are given that
Determine . 
Practice Problem 12J 
Generate an extended truncated negative binomial (ETNB) distribution with and . Note that this is to start with a negative binomial distribution with parameters and and then derive its zerotruncated distribution. The parameters and will not give a distribution but over look this point and go through the process of creating a zerotruncated distribution. In particular, determine the following.

Problem  Answer 

12A 

12B 

12C 

12D 

21E 

12F 

12G 

12H 

12I 

12J 

Dan Ma practice problems
Daniel Ma practice problems
Dan Ma actuarial science
Daniel Ma actuarial science
Dan Ma Math
Dan Ma Mathematics
Daniel Ma Math
Daniel Ma Mathematics
Actuarial exam
2019 – Dan Ma
Practice Problem Set 11 – (a,b,0) class
The practice problems in this post focus on counting distributions that belong to the (a,b,0) class, reinforcing the concepts discussed in this blog post in a companion blog.
The (a,b,1) class is a generalization of (a,b,0) class. It is discussed here. A practice problem set on the (a,b,1) class is found here.
Notation: for where is the counting distribution being focused on.
Practice Problem 11A 
Suppose that claim frequency follows a negative binomial distribution with parameters and . The following is the probability function.
Evaluate the negative binomial distribution in two ways.

Practice Problem 11B 
Suppose that follows a distribution in the (a,b,0) class. You are given that Evaluate the probability that is at least 1. 
Practice Problem 11C 
The following information is given about a distribution from the (a,b,0) class. What is the form of the distribution? Evaluate . 
Practice Problem 11D 
For a distribution from the (a,b,0) class, the following information is given. Determine the variance of this distribution. 
Practice Problem 11E 
For a distribution from the (a,b,0) class, you are given that and . Find the value of . 
Practice Problem 11F 
You are given that the distribution for the claim count satisfies the following recursive relation: Determine . 
Practice Problem 11G 
Suppose that the random variable is from the (a,b,0) class. You are given that and . Calculate the probability that is at least 3. 
Practice Problem 11H 
For a distribution from the (a,b,0) class, you are given that
Determine . 
Practice Problem 11I 
For a distribution from the (a,b,0) class, you are given that and . Evaluate its mean. 
Practice Problem 11J 
The random variable follows a distribution from the (a,b,0) class. You are given that and . Evaluate . 
Practice Problem 11K 
The random variable follows a distribution from the (a,b,0) class. Suppose that and . Determine . 
Practice Problem 11L 
Given that a discrete distribution is a member of the (a,b,0) class. Which of the following statement(s) are true?
B. ……….. 2 only C. ……….. 3 only D. ……….. 1 and 2 only E. ……….. 1 and 3 only 
Practice Problem 11M 
Given that a discrete distribution is a member of the (a,b,0) class, determine the variance of the distribution if and . 
Practice Problem 11N 
For a distribution in the (a,b,0) class, and . Furthermore, the mean of the distribution is 1. Determine . 
Problem  Answer 

11A 

11B 

11C 

11D 

11E 

11F 

11G 

11H 

11I 

11J 

11K 

11L 

11M 

11N 

Daniel Ma Math
Daniel Ma Mathematics
Actuarial exam
2018 – Dan Ma
Practice Problem Set 10 – value at risk and tail value at risk
In actuarial applications, an important focus is on developing loss distributions for insurance products. It is also critical to employ risk measures to evaluate the exposure to risk. This post provides practice problems on two risk measures that are useful from an actuarial perspective. They are: valueatrisk (VaR) and tailvalueatrisk (TVaR).
Practice problems in this post are to reinforce the concepts of VaR and TVaR discussed in this blog post in a companion blog.
Most of the practice problems refer to parametric distributions highlighted in a catalog for continuous parametric models.
Practice Problem 10A 
Losses follow a paralogistic distribution with shape parameter and scale parameter .
Determine the VaR at the security level 99%. 
Practice Problem 10B 
Annual aggregate losses for an insurer follow an exponential distribution with mean 5,000. Evaluate VaR and TVaR for the aggregate losses at the 99% security level. 
Practice Problem 10C 
For a certain line of business for an insurer, the annual losses follow a lognormal distribution with parameters and . Evaluate the valueatrisk and the tailvalueatrisk at the 95% security level. 
Practice Problem 10D 
Annual losses follow a normal distribution with mean 1000 and variance 250,000. Compute the tailvaluerisk at the 95% security level. 
Practice Problem 10E 
An insurance company models its liability insurance business using a Pareto distribution with shape parameter and scale parameter . Evaluate the valueatrisk and the tailvalueatrisk at the 99.5% security level. 
Practice Problem 10F 
Losses follow an inverse exponential distribution with parameter . Calculate the valueatrisk at the 99% security level. 
Practice Problem 10G 
Losses follow a mixture of two exponential distributions with equal weights where one exponential distribution has mean 10 and the other has mean 20. Evaluate the valueatrisk and the tailvalueatrisk at the 95% security level. 
Practice Problem 10H 
Losses follow a mixture of two Pareto distributions with equal weights where one Pareto distribution has shape parameter and scale parameter and the other has shape parameter and scale parameter . Evaluate the valueatrisk and the tailvalueatrisk at the 99% security level. 
Practice Problem 10I 
Losses follow a Weibull distribution with parameters and . Determine the valueatrisk at the security level 99.5%. 
Practice Problem 10J 
Losses follow an inverse Pareto distribution with parameters and . Determine the valueatrisk at the security level 99%. 
Problem  Answer 

10A 

10B 

10C 

10D 

10E 

10F 

10G 

10H 

10I 

10J 

Daniel Ma Math
Daniel Ma Mathematics
Actuarial exam
2018 – Dan Ma
Practice Problem Set 9 – Expected Insurance Payment – Additional Problems
This practice problem set is to reinforce the 3part discussion on insurance payment models (Part 1, Part 2 and Part 3). The practice problems in this post are additional practice problems in addition to Practice Problem Set 7 and Practice Problem Set 8.
Practice Problem 9A 
Losses follow a distribution that is a mixture of two equally weighted Pareto distributions, one with parameters and and the other with with parameters and . An insurance coverage for these losses has an ordinary deductible of 1000. Calculate the expected payment per loss. 
Practice Problem 9B 
Losses, prior to any deductible being applied, follow an exponential distribution with mean 17.5. An insurance coverage has a deductible of 8. Inflation of 15% impacts all claims uniformly from the current year to next year.
Determine the percentage change in the expected claim cost per loss from the current year to next year. 
Practice Problem 9C 
Losses follow a distribution that has the following density function.
An insurance policy is purchased to cover these losses. The policy has a deductible of 3. Calculate the expected insurance payment per payment. 
Practice Problem 9D 
Losses follow a distribution with the following density function.
An insurance coverage pays losses up to a maximum of 100,000. Determine the average payment per loss. 
Practice Problem 9E 
You are given the following information.
Determine the average insurance payment per loss. 
Practice Problem 9F 
You are given the following information.
Determine the proportion of the losses that exceed 1,000. 
Practice Problem 9G 
Losses follow a uniform distribution on the interval . The insurance coverage has a deductible of 250. Determine the variance of the insurance payment per loss. 
Practice Problem 9H 
Losses follow an exponential distribution with mean 500. An insurance coverage that is designed to cover these losses has a deductible of 1,000. Determine the coefficient of variation of the insurance payment per loss. 
Practice Problem 9I 
Losses are modeled by an exponential distribution with mean 3,000. An insurance policy covers these losses according to the following provisions.
Determine the expected insurance payment per loss. 
Practice Problem 9J 
You are given the following information.
Determine the insurance company’s expected claim cost per claim after the effective date of the reinsurance policy. 
Problem  Answer 

9A 

9B 

9C 

9D 

9E 

9F 

9G 

9H 

9I 

9J 

Daniel Ma actuarial
Dan Ma actuarial
2017 – Dan Ma
Practice Problem Set 8 – Expected Insurance Payment – Additional Problems
This practice problem set is to reinforce the 3part discussion on insurance payment models (Part 1, Part 2 and Part 3). The practice problems in this post are additional practice problems in addition to Practice Problem Set 7. Another problem set on expected insurance payment: Practice Problem Set 9.
Practice Problem 8A 
Losses follow a uniform distribution on the interval .
Determine the expected payment per loss. 
Practice Problem 8B 
Losses follow a uniform distribution on the interval .
Determine the expected payment per loss. 
Practice Problem 8C 
Losses in the current year follow a uniform distribution on the interval . Further suppose that inflation of 25% impacts all losses uniformly from the current year to the next year. Losses in the next year are paid according to the following provisions:
Determine the expected payment per loss. 
Practice Problem 8D 
Liability claim sizes follow a Pareto distribution with shape parameter and scale parameter . Suppose that the insurance coverage has a franchise deductible of 20,000 per loss. Given that a loss exceeds the deductible, determine the expected insurance payment. 
Practice Problem 8E 
Losses in the current year follow a Pareto distribution with parameters and . Inflation of 10% is expected to impact these losses in the next year. The coverage for next year’s losses has an ordinary deductible of 1,000.
Determine the expected amount per loss in the next year that will be paid by the insurance coverage. 
Practice Problem 8F 
Losses in the current year follow a Pareto distribution with parameters and . Inflation of 10% is expected to impact these losses in the next year. The coverage for next year’s losses has a franchise deductible of 1,000. Determine the expected amount per loss in the next year that will be paid by the insurance coverage. 
Practice Problem 8G 
Losses follow a distribution that is a mixture of two equally weighted exponential distributions, one with mean 6 and the other with mean 12. An insurance coverage for these losses has an ordinary deductible of 2. Calculate the expected payment per loss. 
Practice Problem 8H 
Losses follow a distribution that is a mixture of two equally weighted exponential distributions, one with mean 6 and the other with mean 12. An insurance coverage for these losses has a franchise deductible of 2. Calculate the expected payment per loss. 
Practice Problem 8I 
You are given the following information.
Determine the average payment per loss. 
Practice Problem 8J 
You are given the following information.
Determine the percentage change in the expected claim cost per loss when losses are uniformly impacted by a 20% inflation. 
All normal probabilities are obtained by using the normal distribution table found here.
Problem  Answer 

8A 

8B 

8C 

8D 

8E 

8F 

8G 

8H 

8I 

8J 

Daniel Ma actuarial
Dan Ma actuarial
2017 – Dan Ma
Practice Problem Set 7 – Expected Insurance Payment
This practice problem set is to reinforce the 3part discussion on insurance payment models (Part 1, Part 2 and Part 3). The practice problems in this post are basic problems on calculating average insurance payment (per loss or per payment).
Additional problem set: Practice Problem Set 8 and Practice Problem Set 9.
Practice Problem 7A 
Losses follow a uniform distribution on the interval .

Practice Problem 7B 
Losses for the current year follow a uniform distribution on the interval . Further suppose that inflation of 25% impacts all losses uniformly from the current year to the next year.

Practice Problem 7C 
Losses follow an exponential distribution with mean 5,000. An insurance policy covers losses subject to a franchise deductible of 2,000. Determine the expected insurance payment per loss. 
Practice Problem 7D 
Liability claim sizes follow a Pareto distribution with shape parameter and scale parameter . Suppose that the insurance coverage pays claims subject to an ordinary deductible of 20,000 per loss. Given that a loss exceeds the deductible, determine the expected insurance payment. 
Practice Problem 7E 
Losses follow a lognormal distribution with and . For losses below 1,000, no payment is made. For losses exceeding 1,000, the amount in excess of the deductible is paid by the insurer. Determine the expected insurance payment per loss. 
Practice Problem 7F 
Losses in the current exposure period follow a lognormal distribution with and . Losses in the next exposure period are expected to experience 12% inflation over the current year. Determine the expected insurance payment per loss if the insurance contract has an ordinary deductible of 1,000. 
Practice Problem 7G 
Losses follow an exponential distribution with mean 2,500. An insurance contract will pay the amount of each claim in excess of a deductible of 750. Determine the standard deviation of the insurance payment for one claim such that a claim includes the possibility that the amount paid is zero. 
Practice Problem 7H 
Liability losses for auto insurance policies follow a Pareto distribution with and . These insurance policies have an ordinary deductible of 1,250. Determine the expected payment made by these insurance policies per loss. 
Practice Problem 7I 
Liability losses for auto insurance policies follow a Pareto distribution with and . These insurance policies make no payment for any loss below 1,250. For any loss greater than 1,250, the insurance policies pay the loss amount in excess of 1,250 up to a limit of 5,000. Determine the expected payment made by these insurance policies per loss. 
Practice Problem 7J 
Losses follow a lognormal distribution with and . For losses below 1,000, no payment is made. For losses exceeding 1,000, the amount in excess of the deductible is paid by the insurer. Determine the average insurance payment for all the losses that exceed 1,000. 
All normal probabilities are obtained by using the normal distribution table found here.
Problem  Answer 

7A 

7B 

7C 

7D 

7E 

7F 

7G 

7H 

7I 

7J 

Daniel Ma actuarial
Dan Ma actuarial
2017 – Dan Ma
Practice Problem Set 6 – Negative Binomial Distribution
This post has exercises on negative binomial distributions, reinforcing concepts discussed in
this previous post. There are several versions of the negative binomial distribution. The exercises are to reinforce the thought process on how to use the versions of negative binomial distribution as well as other distributional quantities.
Practice Problem 6A 
The annual claim frequency for an insured from a large population of insured individuals is modeled by the following probability function.
Determine the following:

Practice Problem 6B 
The number of claims in a year for an insured from a large group of insureds is modeled by the following model. The parameter varies from insured to insured. However, it is known that is modeled by the following density function. Given that a randomly selected insured has at least one claim, determine the probability that the insured has more than one claim. 
Practice Problem 6C 
Suppose that the number of accidents per year per driver in a large group of insured drivers follows a Poisson distribution with mean . The parameter follows a gamma distribution with mean 0.6 and variance 0.24. Determine the probability that a randomly selected driver from this group will have no more than 2 accidents next year. 
Practice Problem 6D 
Suppose that the random variable follows a negative binomial distribution such that
Determine the mean and variance of . 
Practice Problem 6E 
Suppose that the random variable follows a negative binomial distribution with mean 0.36 and variance 1.44.
Determine . 
Practice Problem 6F 
A large group of insured drivers is divided into two classes – “good” drivers and “bad”drivers. Seventy five percent of the drivers are considered “good” drivers and the remaining 25% are considered “bad”drivers. The number of claims in a year for a “good” driver is modeled by a negative binomial distribution with mean 0.5 and variance 0.625. On the other hand, the number of claims in a year for a “bad” driver is modeled by a negative binomial distribution with mean 2 and variance 4. For a randomly selected driver from this large group, determine the probability that the driver will have 3 claims in the next year. 
Practice Problem 6G 
The number of losses in a year for one insurance policy is the random variable where . The random variable is modeled by a geometric distribution with mean 0.4 and variance 0.56.
What is the probability that the total number of losses in a year for three randomly selected insurance policies is 2 or 3? 
Practice Problem 6H 
The random variable follows a negative binomial distribution. The following gives further information.
Determine and . 
Practice Problem 6I 
Coin 1 is an unbiased coin, i.e. when tossing the coin, the probability of getting a head is 0.5. Coin 2 is a biased coin such that when tossing the coin, the probability of getting a head is 0.6. One of the coins is chosen at random. Then the chosen coin is tossed repeatedly until a head is obtained.
Suppose that the first head is observed in the fifth toss. Determine the probability that the chosen coin is Coin 2. 
Practice Problem 6J 
In a production process, the probability of manufacturing a defective rear view mirror for a car is 0.075. Assume that the quality status of any rear view mirror produced in this process is independent of the status of any other rear view mirror. A quality control inspector is to examine rear view mirrors one at a time to obtain three defective mirrors.
Determine the probability that the third defective mirror is the 10th mirror examined. 
Problem  Answer 

6A 

6B  
6C  0.9548 
6D  mean = 0.65, variance = 0.975 
6E  0.016963696 
6F  0.04661 
6G  
6H 

6I  0.329543 
6J  0.008799914 
Daniel Ma Math
Daniel Ma Mathematics
Actuarial exam
2017 – Dan Ma