Assignment:
1. Let X(t) be a Poisson process with parameter λ = 3 arrival/s. Find the probability of having 2 arrivals in the time interval (0, 1], given that there are 5 arrivals in the time interval (0, 3].
2. Consider a system where items arrive according to a Poisson process with parameter λ. Items are collected in a storage facility that can host up to Q objects. When the storage facility is full, that is, exactly Q items have been collected, the system empties its storage facility in zero time and starts collecting items anew. Let N(t) be the number of items in the system at time t, with N(0) = 0. Furthermore, let T = min{t ≥ 0 : N(t) = Q} the time at which the storage facility fills up for the first time. Prove that
E|T|= Q/λ And E [T∫0 N(t)dt] = 1+2+...+Q-1/λ= Q(Q-1)/2λ