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ACTSC 432 (1)

Test 1 - S12.pdf

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University of Waterloo
Actuarial Science
Diana Parry

ACTSC 432/832 - TEST #1 Name : ID Number : 1. (10 marks) Suppose that, conditional on ▯ = ▯, the rv’ s fX j are independent and identically j▯1 distributed with common probability mass function ▯ e▯▯ pXj▯ (xj▯) = , x = 0;1;::: x! We further assume that ▯ is a rv with the following two-point mixture density function ▯ ▯▯▯ ▯ 2 ▯▯▯ ▯▯(▯) = p ▯e + (1 ▯ p) ▯ ▯e , ▯ > 0. (a) (3 marks) …nd E [X ]. j 1 (b) (4 marks) determine V ar (X ). j 2 (c) (3 marks) …nd Cov (X ;X ) for i 6=jj. 3 2. (12 marks) Let X ;X 1:::2X be tne past total claims experience for a given policyholder. We assume that the X ’is are independent and identically distributed (iid) rs. More precisely, the X is are distributed as a rv S which is de…ned as follows: ▯ S j▯ = ▯ is a compound Poisson rv, i.e. ▯ P N S = j=1Y j N > 0, 0, N = 0, where N j▯ = ▯ is Poisson distributed with mean ▯, and the Y ’ js are a sequence of iid r.v.s (also independent of N and ▯) with a mean equals to 2 times its standard deviation. ▯ ▯ is a positive rv with V ar (▯) = 3E [▯]. Under the number of claims basis, the American credibility factor Z was found to be 0:45. All else being equal, …nd the American credibility factor on the total amount of claims basis.
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