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IAI Actuarial Core Principles · Risk Modelling and Survival Analysis · Compound distributions and their applications in risk modelling

Claim numbers N follow a binomial distribution with n = 3 and p = 0.4. Claim amounts are always 1 (in rupee lakhs). Using the (a,b,0) recursion for aggregate claims S, which values of a and b apply, and what is P(S = 2)?

For a binomial with n=3 and p=0.4, a = -2/3 and b = 8/3. Since each claim is 1, S equals N and P(S=2) = 3(0.4)^2(0.6) = 0.288.

  1. Aa = -2/3, b = 2.0; P(S=2) = 0.3456Correct
  2. Ba = -0.4, b = 1.6; P(S=2) = 0.2880
  3. Ca = 0.4, b = 1.2; P(S=2) = 0.3456
  4. Da = -2/3, b = 2.0; P(S=2) = 0.2880
  5. a = -0.6, b = 2.4; P(S=2) = 0.1440

Explanation

Binomial: a = -p/(1-p) = -0.4/0.6 = -2/3 and b = (n+1)p/(1-p) = 4*0.4/0.6 = 8/3. Check: p1/p0 = a+b = 2 versus 3*0.4/0.6 = 2, so b = 8/3 and a+b = 2. The listed b=2.0 is wrong, so no option is fully correct; the true P(S=2)=P(N=2)=3*0.16*0.6=0.288.

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