ACTSC 432/832 - TEST #1
ID Number :
1. (10 marks) Suppose that, conditional on ▯ = ▯, the rv s fX j are independent and identically
distributed with common probability mass function
pXj▯ (xj▯) = , x = 0;1;:::
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 ].
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.