Q3.

3. Prove that the following processes are Poisson with rates as illustrated in the figures a) The merged process (union of event times). b) The split process (events split according to the time-invariant p.m.f. 1)). Further show that the split processes are independent from one another. - Merging - Splitting I Union of event times - Classify events with prob. pl- 31 '11 = 131}. n 12 = 1192/1 '12 Z a\"! A : 5 i=1 I in in = pnl In addition, show that the merged process has the same distribution as the sum process L1 Ni(t) (where {Ni(t)} are independent Poissonai) processes). Why is it possible that the sum process and merged process may have different sample paths (but this happens with zero probability)