Consider a Poisson process with rate lambda=1. Consider three times that satisfy 0<t1<t2<t3. Let M be the number of arrivals during the interval [0,t2]. Let N be the number of arrivals during the interval [t1+ t3]. Is the random variable M+N guaranteed to be Poisson?
Yes, the sum of two Poisson random variables with rates lambda1 and lambda2 is also a Poisson random variable with rate lambda1 + lambda2.
In this case, M is the number of arrivals during the interval [0,t2] which has a Poisson distribution with rate lambda*t2. Similarly, N is the number of arrivals during the interval [t1, t3] which has a Poisson distribution with rate lambda*(t3-t1).
Therefore, M + N is the sum of two Poisson random variables with rates lambda*t2 and lambda*(t3-t1). Thus, the random variable M+N is guaranteed to be Poisson with rate lambda*(t2+t3-t1).