# Severity Models

### Practice Problem Set 5 – Exercises for Severity Models

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This problem set has exercises to reinforce the various parametric continuous probability models discussed in the companion blog on actuarial modeling. Links are given below for the models involved.

This blog post in Topics in Actuarial Modeling has a catalog for continuous models.

 Practice Problem 5A Claim amounts for collision damages to insured cars are mutually independent random variables with common density probability density function $\displaystyle f(x)=\left\{ \begin{array}{ll} \displaystyle \frac{1}{1500} \ e^{-x/1500} &\ X > 0 \\ \text{ } & \text{ } \\ \displaystyle 0 &\ \text{otherwise} \end{array} \right.$ For three claims that are expected to be made, calculate the expected value of the largest of the three claims.
 Practice Problem 5B The lifetime of an electronic device is modeled using the random variable $Y=20 X^{3}$ where $X$ is an exponential random variable with mean 0.5. Determine the variance of $Y$.
 Practice Problem 5C The lifetime (in years) of an electronic device is $2X +Y$ where $X$ and $Y$ are independent exponentially distributed random variables with mean 3.5. Determine the probability density function of the lifetime of the electronic device.
 Practice Problem 5D The time (in years) until the failure of a machine that is brand new is modeled by a Weibull distribution with shape parameter 1.5 and scale parameter 4. Calculate the 95th percentile of times to failure of the machines that are 2-year old.
 Practice Problem 5E The size of a bodily injury claim for an auto insurance policy follows a Pareto Type II Lomax distribution with shape parameter 2.28 and scale parameter 200. Calculate the proportion of claims that are within one-fourth standard deviations of the mean claim size.
 Practice Problem 5F Suppose that the size of a claim has the following density function. $\displaystyle f(x)=\left\{ \begin{array}{ll} \displaystyle \frac{1}{2.5 \ x \ \sqrt{2 \pi}} \ e^{- z^2/2} &\ x > 0 \\ \text{ } & \text{ } \\ \displaystyle 0 &\ \text{otherwise} \end{array} \right.$ where $\displaystyle z=\frac{\text{ln}(x)-1.1}{2.5}$. A coverage pays claims subject to an ordinary deductible of 20. Determine the expected amount paid by the coverage per claim.
 Practice Problem 5G An actuary determines that sizes of claims from a large portfolio of insureds are exponentially distributed. For about 60% of the claims, the claim sizes are modeled by an exponential distribution with mean 1.2. For about 30% of the claims, the claim sizes are modeled by an exponential distribution with mean 2.8. For the remaining 10% of the claims, the claim sizes are considered high claim sizes and are modeled by an exponential distribution with mean 7.5. Determine the variance of the size of a claim that is randomly selected from this portfolio.
 Practice Problem 5H Losses are modeled by a loglosistic distribution with shape parameter $\gamma=2$ and scale parameter $\theta=10$. When a loss occurs, an insurance policy reimburses the loss in excess of a deductible of 5. Determine the 75th percentile of the insurance company reimbursements over all losses.
 Practice Problem 5I Losses are modeled by a loglosistic distribution with shape parameter $\gamma=2$ and scale parameter $\theta=10$. When a loss occurs, an insurance policy reimburses the loss in excess of a deductible of 5. Determine the 75th percentile of the insurance company reimbursements over all payments.
 Practice Problem 5J Claim sizes for a certain class of auto accidents are modeled by a uniform distribution on the interval $(0, 10)$. Five accidents are randomly selected. Determine the expected value of the median of the five accident claims.

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Problem Links for the relevant distributions
5A Exponential distribution
5B Exponential distribution
5C Hypoexponential distribution
5D Weibull distribution
5E Pareto distribution
5F Lognormal distribution and limited expectation
5G Hyperexponential distribution
5H Loglogistic distribution
5I Loglogistic distribution
5J Order statistics

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5A 2,750
5B 4,275
5C $\displaystyle \frac{2}{7} \ e^{-x/7}-\frac{1}{3.5} \ e^{-x/3.5}$
5D 8.954227
5E 0.4867
5F 61.106
5G 12.3459
5H 12.32
5I 15
5J 5

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