1. Consider a Poisson process with intensity λ. We start observing at time t = 0. Let T be the time that has elapsed at the first occurrence. Continue to observe the process T further units of time. Let N (T ) be the number of occurrences during the latter period (i.e., during (T, 2T ]). Determine the distribution of N (T).
2. A particle source A emits one particle at a time, according to a Poisson process with an intensity of two particles a minute. Another particle source B emits two particles at a time, according to a Poisson process with an intensity of one pair of particles a minute. The sources are independent of each other. We begin to observe the sources at time zero. Compute the probability that source A has emitted two particles before source B has done so.