IAI Actuarial Core Principles · Risk Modelling and Survival Analysis · Stochastic processes
A Poisson process has rate λ = 2 per day. Let T3 be the time of the third event. What are E[T3] and Var[T3]?
E[T3] is 1.5 days and Var[T3] is 0.75 days squared. The third event time is a sum of three independent exponential waiting times with rate 2, so it is Gamma(3, 2), with mean 3/2 and variance 3/4.
- A1.5 days and 0.75 days²Correct
- B1.5 days and 1.5 days²
- C6 days and 12 days²
- D0.5 days and 0.25 days²
- 1.5 days and 0.375 days²
Explanation
T3 is the sum of 3 independent exponential(2) inter-event times, so it is Gamma(3, 2). Mean = 3/2 = 1.5. Variance = 3/2² = 0.75. Option 1 uses the mean as the variance.
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