Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit

MTH 223: Mathematical Risk Theory

Tutorial 2

1. Let X ∼ PAR(α, θ) with parameters α > 0, θ > 0, and df X(x) = , x > 0. Determine whether X has a increasing failure rate func-tion or decreasing failure rate function distribution. Is it a heavy-tailed or light-tailed distribution?

2. Compare the tail weight of the Weibull and the inverse Weibull distri-butions by the following criteria respectively. You may use some facts with respect to these two distributions from the Distribution Table posted on course webpage.

(a) Existence of moments

(b) Ratio of density functions

3. You are given that the random variable X has pdf

(a) Determine the survival function (x).

(b) Determine the hazard rate function h(x).

(c) Determine the mean residual life function e(x).

(d) Determine limx→∞ h(x) and limx→∞ e(x).

(e) Prove that e(x) is strictly decreasing but h(x) is not strictly in-creasing.

4. The conditional hazard rate function of loss X, given Λ = λ, is h(x | λ) = λx3 . Λ has a gamma distribution GAM(2, 4).

(a) Calculate the probability that the loss is less than one.

(b) Is the distribution of X decreasing failure rate function, increasing failure rate function, or neither? Justify your answer.

5. An inverse Gaussian distribution has a pdf as follows

Show that the inverse Gaussian distribution as parameterized above is a scale family but does not have a scale parameter.

6. A Weibull distribution has a pdf

Show that the Weibull distribution has a scale parameter.