1.4 Sum of Poisson 1. Let X and Y be independent r.v.s, both are Poisson distributed with parameter respectively A and . Compute the distribution of Z = X + Y. We can recall Newton's formula: ( a + b ) " = > k=0 2. Consider two independent Poisson point processes (N.")(20 and (N." )(20, respec- tively with rate A and #. In particular, for any s, t, N," and N." are indepen dent, and both are Poisson distributed with parameter As and ut respectively. Let M = N" + N," for all t > 0. (a) What is the distribution of Nt for t 2 0. (b) What is the distribution of N([a, b]), where N([a, b] ) = No - Na. Does (Ni)+20 verify the homogeneous assumption of Poisson point process? Let's recall that, the homogeneous assumption is to say, in any time interval, the average number of points is proportional to the length of the interval. (c) Consider a large insurance company, dealing with house insurance as well as car insurance. N' are the Poisson point process associated to house accidentclaims, N(3) are the Poisson point process associated to car accident claims, what do you say of the process N.? Is it a Poisson point process? This is a yes-or-no