ASSIGNMENT 3 – ACTSC 431/831, FALL 2008 Due at the beginning of the class on Tuesday, October 21 . 1. Let loss X have a three-component spliced distribution. The pdf of X has the following form: f X ( x ) = 0 , x <0 ,0 . 06 ,0 ≤ x < 5 ,0 . 1 , 5 ≤ x < 10 , a x 4 , x ≥ 10 . (a) Calculate the variance of the loss. [6 marks] (b) Calculate the distribution function F X ( x ) of the loss for all x ∈ (-∞ , ∞ ). [6 marks] (c) Calculate the probability that the loss is between 4 and 8. [2 marks] 2. Let N be the number of claims in an insurance portfolio. The sizes of claims in the portfolio are i.i.d. random variables with common Pareto (3 , 50) distribution. Furthermore, the sizes of claims are independent of N . Let N 1 be the number of claims of sizes between 5 and 10 and N 2 be the number of claims of sizes larger than their expectation. (a) Assume that
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