DUE DATE: Friday, February 10, 2023

INSTRUCTIONS: Students may discuss material covered in class with their classmates. However, under no circumstances should you share your homework file with anyone. Use R code wherever appropriate to do numerical work, but make sure you include text to explain your computations where relevant.

Submit your homework on the Canvas site as a PDF file called hw1-yournetid.pdf.     You may write up your homework using R Markdown by editing the .Rmd file provided. Here is a link to a Markdown Basics page.

Unless indicated otherwise all parts of all questions are worth 1 point, for a total of 15.

1. Figure 3.7 in the Loss Models text plots the Pareto density with parameters α = 3 and θ = 10, and the the gamma density with parameters α = 1/3 and θ = 15.

1a. Show that the two distributions have the same means and variances.

1b. Plot the Pareto density function over the range (50, 150).

1c. Overlay the gamma density.

Use different colors for the two distributions and the legend command to indicate which is which.

2. Let S (x; α, θ) denote the survival function for a gamma distribution with shape α and scale θ . 2a. Show that S (x) = e−x/3/4 + 3S(x; 4, 2)/4, x ≥ 0, is a survival function.

2b. Determine the density function corresponding to the survival function in part 2a.

2c. Use L’Hopital’s rule to determine the limiting value of the hazard function.

2d. Explain why the distribution might be considered light-tailed.

3. Consider the gamma and Pareto distributions in problem 1.

3a. Plot the hazard functions of the two distributions over the range (50, 150). Include horizontal (dashed) lines at the limiting values.

3b.  (2 points) Plot the mean residual life distributions over the range (50, 150).  (You will need to use a numerical integration routine such as integrate to do this for the gamma distribution.)

Use different colors for the two distributions and the legend command to indicate which is which.

4. Suppose that X has a Pareto distribution with parameters α and θ . Suppose that an insurance policy has a loss distribution given by Y = min(X, u), where u > 0 is a policy limit.

4a. (2 points) Derive a formula for the expected loss. (Hint: See eq:(3.9) in the Loss Models book.)        4b. Tabulate the expected loss if α = 2.5, θ = 150 for u = 500, 1000, 2000 and if there is no policy limit.