Question: 2. (a) Let N(t), t> 0 be a Poisson process with rate X. State without proof which of the following is true for positive h

2. (a) Let N(t), t> 0 be a Poisson process with

2. (a) Let N(t), t> 0 be a Poisson process with rate X. State without proof which of the following is true for positive h as h0: (i) P(N(h) = 0) = o(h) [3] (ii) P(N(h) = 1) = 1h + oh) [2] (iii) P(N(h) = 2) = o(h) [2] (iv) E[N(h)] = 1h + och) [3] (b) Suppose that a hunter shoots the ducks according to a Poisson process with rate 3 per hour. He starts hunting at 8 a.m. (i) What is the probability that the hunter shot exactly 6 ducks by 10 a.m., 8 ducks by 11 a.m. and 14 ducks by 1 p.m.? [2] (ii) What is the expectation of the time the hunter needs to shoot 6 ducks? [2] (iii) Suppose that we know that by 10 a.m. the hunter shot exactly 7 ducks. What is the probability that by 1 p.m. the hunter shot exactly 16 ducks? [2] (iv) Suppose that we know that by 11 p.m. a hunter shot exactly 9 ducks. What is the probability that he shot his first duck after 9 a.m.? [2] (v) Suppose the hunter did not shoot any duck before 10 a.m. What is the probability that he shot his first duck after 11 a.m.? [2]

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