Hi Please explain that in Q 4.1 ii) how will we denote the required probability in terms of T and N. Is it P(Nt=1)? Not too clear with this concept. Thank you.
Hello \( P(N_t = 1) = P(N_t - N_0 = 1) \) is a special case of the probability. More generally, the question is asking you to work out: \( P(N_{s + t} - N_s = 1) \) for any s >= 0, t > 0 ie for any interval of length t (so from some time s to time s + t) the probably of seeing one arrival in that interval. Due to the nature of a Poisson process, this probability is the same for any s>=0 for some fixed t >0. For a Poisson process, the distribution of \( N_{s + t} - N_s \) is \( Poi(\lambda t) \) for any s >= 0, t > 0. Hope this helps Andy
