Reference no: EM13494424
Exercise 1. In particular country, storms are quite frequent any occur, on average, 15 storms every year. Their occurence times can be assumed to follow a marked Poisson process, where the marks are the wind speed during the storm. The speed (in kph) of the storms has an exponential distribution with parameter θ =1/60. The storm induces property damage when the speed is > 80.
1) What is the probability that there is no storm in january ?
2) What is the probability that there is no damage-inducing storm in january ?
3) What is the expectation of the number of damage-inducing storms ?
4) Given that there was 6 storms last summer, what is the probability that at least 4 of them caused property damage ?
Exercise 2. Let {Xt}t≥0 be a homogeneous Poisson process with intensity parameter λ. Suppose that we observe this process only when the time t is an integer. Therefore this de?nes a discrete time process {Xn}nεN. Prove that this process satis?es the Markov property.
Exercise 3. Let {Xt} be a inhomogeneous Poisson process with intensity λ(t)= 1 + cos(Πt).
1) Calculate P(X3 = 2).
2) Prove that P(X1+h - X1 > 0) = h3Π3/6 = o(h3)
Exercise 4. Let {Xt}t≥0 be a inhomogeneous Poisson process with intensity λ(t)= at for an unknown parameter a> 0. One observes Xt for 0 ≤ t ≤ T. Propose an estimator ^aT of a and state convergence results for this estimator.
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What is the probability that there is no storm in january
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