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IAI Actuarial Core Principles · Risk Modelling and Survival Analysis · Stochastic processes

Claims follow a Poisson process with rate 3 per month. Each claim is independently a large claim with probability 0.2. Let X be the number of large claims in 2 months and Y the number of other claims in 2 months. Which statement is correct?

X is Poisson with mean 1.2, Y is Poisson with mean 4.8, and they are independent. The total mean over 2 months is 6, and random independent splitting of a Poisson process gives independent Poisson processes with the thinned rates.

  1. AX is Poisson with mean 1.2, Y is Poisson with mean 4.8, and X and Y are independentCorrect
  2. BX is Poisson with mean 1.2, Y is Poisson with mean 4.8, and X and Y are negatively correlated
  3. CX is Poisson with mean 0.6, Y is Poisson with mean 2.4, and X and Y are independent
  4. DX is binomial with mean 1.2, Y is Poisson with mean 4.8, and X and Y are independent
  5. X is Poisson with mean 1.2, Y is Poisson with mean 6, and X and Y are independent

Explanation

Total mean in 2 months is 6. Thinning gives large claims mean 6 x 0.2 = 1.2 and other claims mean 6 x 0.8 = 4.8. By the thinning property, the two processes are independent Poisson. Negative correlation would arise only if the total were fixed.

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