1.4 Sum of Poisson 1. Let X and Y be independent r.v.s, both are Poisson distributed with parameter respectively and M. Compute the distribution of Z = X +Y. We can recall Newton's formula: " n n (a+b)* = ()* (1) a -k k=0 2. Consider two independent Poisson point processes (N_C")zo and (N2))>0, respec- tively with rate , and pl. In particular, for any s, t, ns") and N, 2) are indepen- dent, and both are Poisson distributed with parameter Is and ut respectively. Let Ne = N(1) + N(2) for all t > 0. (a) What is the distribution of N4 for t > 0. (b) What is the distribution of N([a,b]), where N([a, b]) = No Na. Does (Nt)t>0 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{1) are the Poisson point process associated to house accident = 1.4 Sum of Poisson 1. Let X and Y be independent r.v.s, both are Poisson distributed with parameter respectively and M. Compute the distribution of Z = X +Y. We can recall Newton's formula: " n n (a+b)* = ()* (1) a -k k=0 2. Consider two independent Poisson point processes (N_C")zo and (N2))>0, respec- tively with rate , and pl. In particular, for any s, t, ns") and N, 2) are indepen- dent, and both are Poisson distributed with parameter Is and ut respectively. Let Ne = N(1) + N(2) for all t > 0. (a) What is the distribution of N4 for t > 0. (b) What is the distribution of N([a,b]), where N([a, b]) = No Na. Does (Nt)t>0 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{1) are the Poisson point process associated to house accident =