Severity Guided Tutorials

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Here are a set of exercises that guide the viewer through some of the theoretical foundations of Loss Data Analytics. Each tutorial is based on one or more questions from the professional actuarial examinations – typically the Society of Actuaries Exam C.

Tutorial Structure. Each guided tutorial has a strategy set that describes the context. When you hit the “Start quiz” button, you begin the tutorial that is comprised of a series of mini-questions designed to lead you to the target question. At each stage, hints are provided as well as feedback on the correct solution of each mini-question.

Your Assignment. In reviewing these exercises, ideally the viewer will:

  • Work the problem posed referring only to basic theory
  • Even if you get the answer correct, review the strategy for this type of problem by clicking (revealing) the Strategy for … header
  • If you feel comfortable with the strategy and got the problem correct, then you may choose to move on. However, you might also decide to follow the step-by-step process for solving the problem by clicking on the “Start Quiz” button. It is not really a quiz — it is a guided tutorial.

Strategy for Mixture Problems

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Two-Point Mixture 169
The distribution of a loss, X, is a two-point mixture:
(i) With probability 0.8, X has a two-parameter Pareto distribution with (alpha=2) and (theta=100).
(ii) With probability 0.2, X has a two-parameter Pareto distribution with with (alpha=4) and (theta=3000).
Calculate Pr((Xleq200).)

[WpProQuiz 60]


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Strategy for Obtaining Maximum Likelihood Estimators for Severity Problems


Pareto Likelihood 37
A random sample of three claims from a dental insurance plan is given below:
$$
{scriptsize
begin{matrix}
begin{array}{ccc}
225 & 525 & 950 \
end{array}
end{matrix}
}
$$Claims are assumed to follow a Pareto distribution with parameters (theta=150) and (alpha).

Calculate the maximum likelihood estimate of (alpha).


[WpProQuiz 62]


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Strategy for Obtaining Maximum Likelihood Estimators for Mixture Problems


Mixture Distribution Likelihood Estimation 26
You are given:
(i) Low-hazard risks have an exponential claim size distribution with mean (theta).
(ii) Medium-hazard risks have an exponential claim size distribution with mean (2theta).
(iii) High-hazard risks have an exponential claim size distribution with mean (3theta).
(iv) No claims from low-hazard risks are observed.
(v) Three claims from medium-hazard risks are observed, of sizes 1, 2 and 3.
(vi) One claim from a high-hazard risk is observed, of size 15.

Calculate the maximum likelihood estimate of (theta)


[WpProQuiz 63]


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Strategy for Likelihood Ratio Test Problems


Pareto Likelihood 22
You fit a Pareto distribution to a sample of 200 claim amounts and use the likelihood ratio test to test the hypothesis that (alpha=1.5 ) and (theta=7.8 )
You are given:
(i) The maximum likelihood estimates are (widehat{alpha}=1.4) and (widehat{theta}=7.6)
(ii) The natural logarithm of the likelihood function evaluated at the maximum likelihood estimates is (-817.92)
(iii) (sumln(x_i +7.8)=607.64)

Determine the result of the test.


[WpProQuiz 61]

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Strategy for Loss Elimination Ratio Problems


Loss Elimination Ratio 89
You are given:
(i) Losses follow an exponential distribution with the same mean in all years.
(ii) The loss elimination ratio this year is 70%.
(iii) The ordinary deductible for the coming year is 4/3 of the current deductible.

Calculate the loss elimination ratio for the coming year.


[WpProQuiz 65]


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Strategy for Simulation using Inverse Transform Method Problems


Simulation 202
Unlimited claim severities for a warranty product follow the lognormal distribution with parameters (mu=5.6) and (sigma =0.75). You use simulation to generate severities. The following are six uniform (0, 1) random numbers:
$$
{scriptsize
begin{matrix}
begin{array}{ccc}
0.6179 & 0.4602 & 0.9452 & 0.0808 & 0.7881 & 0.4207\
end{array}
end{matrix}
}
$$ Using these numbers and the inversion method, calculate the average payment per claim for a contract with a policy limit of 400.

[WpProQuiz 66]


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Strategy for Two-Point Mixture Simulation Problems


Mixture Simulation 290
A random variable X has a two-point mixture distribution with pdf
$$ f(x) = frac{1}{8} e^{-x/2} + frac{1}{4} e^{-x/3}.$$You are to simulate one value, x, from this distribution using uniform random numbers 0.2 and 0.6. Use the value 0.2 and the inversion method to simulate J where J=1 refers to the first random variable in the mixture and J=2 refers to the second random variable. Then use 0.6 and the inversion method to simulate a value from X.
Calculate the value of x.

[WpProQuiz 67]


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Continuous Mixture 204
The length of time, in years, that a person will remember an actuarial statistic is modeled by an exponential distribution with mean 1/Y. In a certain population, Y has a gamma distribution with (alpha)=(theta)=2.

Calculate the probability that a person drawn at random from this population will remember an actuarial statistic less than 1/2 year.


[WpProQuiz 59]

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