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 
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