This practice problem set has more exercises on maximum likelihood estimation, the continuation of Practice Problem Set 2.
This problem set and the previous one present basic practice problems to reinforce the concepts discussed in two posts – this one and this one. The first post shows how to obtain maximum likelihood estimates given complete data (individual data). The second post focuses on maximum likelihood estimation for other data scenarios (grouped data, censored data and truncated data).
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Practice Problem 3A 
The following information is given about a sample of 5 observations.
Determine the following:

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Practice Problem 3B 
A random sample of 10 losses are given below:
The distribution that models the losses is known to be an exponential distribution with mean .

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Practice Problem 3C 
In a study of patients with cardiovascular disease, 5 patients are observed for a period 5 years. Three of the patients die during the study with their times of death at 1, 1, 3. The remaining two patients survive to the end of the study. The time until death is modeled by a distribution with the following cumulative distribution function:

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Practice Problem 3D 
An insurance coverage has a deductible of 50. A sample of 7 losses is given: 65, 100, 150, 200, 350, 505 and 600. No information is known about losses below 50. The losses are known to follow an exponential distribution with mean .

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Practice Problem 3E 
An insurance coverage has a deductible of 50 and a maximum covered loss of 500. A sample of 7 losses is given: 65, 100, 150, 200, 350, 500 and 500. No information is known about losses below 50. The two data points of 500 are the result of censoring at 500. The losses are known to follow an exponential distribution with mean .

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Practice Problem 3F 
An insurance coverage has a deductible of 50 and a maximum covered loss of 500. A sample of 7 payments is given: 15, 50, 100, 150, 300, 450 and 450. The two data points of 450 are the result of censoring at 500 and then subtracting the deductible. The payments are known to follow an exponential distribution with mean .

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Practice Problem 3G 
An insurance coverage has a deductible of 50 and a maximum covered loss of 500. A sample of 7 losses is given: 55, 60, 100, 150, 250, 500 and 500. No information is known about losses below 50. The two data points of 500 are the result of censoring at 500. The losses (including the losses below the deductible and the losses exceeding the limit) are known to follow a Pareto distribution with parameters and .

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Practice Problem 3H 
An insurance coverage has a deductible of 5. The following claims are observed:
The above sample is the result of a truncation below at 5. A Weibull distribution with parameters and is fitted to the loss distribution (including losses below the deductible).

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Practice Problem 3I 
Two groups of insureds are pooled for maximum likelihood estimation. Losses for Group 1 has a Pareto distribution with parameters and . Losses for Group 2 has a Pareto distribution with parameters and . The following losses have been observed:
Group 2: 875, 980, 1500

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Practice Problem 3J 
Suppose that the lifetimes of a certain type of washing machines have a Weibull distribution with parameters and . Seven such machines are tested during a 5year period. Two of the machines fail before the end of the testing period. Their times at failure are 2, 3. The other 5 machines are in working condition at the end of the testing period.

Problem  Answer 

3A 

3B 

3C 

3D 

3E 

3F 

3G 

3H 

3I 

3J 

actuarial practice problems
Dan Ma actuarial
Daniel Ma actuarial
Daniel Ma Math
Daniel Ma Mathematics
Actuarial exam
2018 – Dan Ma
Tagged: Maximum Likelihood Estimation, Maximum Likelihood Estimators
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