1. Consider a sender that transmits signals according to a Poisson process with intensity λ. The signals are received by a receiver, however, in such a way that every signal is registered with probability p, 0 p 1, and "missed" with probability q = 1 - p. Registrations are independent. Let X(t) be the number of transmitted signals during (0, t], let Y (t) be the number of registered signals, and let Z(t) be the number of nonregistered signals during this period, wheret ≥ 0.
(a) Show that Y (t) and Z(t) are independent, and determine their distri- butions.
(b) Determine the distribution of the number of signals that have been transmitted when the first signal is registered.
(c) Determine the distribution of the number of signals that have been transmitted when the kth signal is registered.
(d) Determine the conditional distribution of the number of registered signals given the number of transmitted signals, that is, compute P (Y (t) = k | X(t) = n) for suitable choices of k and n.
(e) Determine the conditional distribution of the number of transmit- ted signals given the number of registered signals, that is, compute P (X(t) = n | Y (t) = k) for suitable choices of k and n. Remark. It thus follows from (a) that the number of registered signals during a given time period provides no information about the actual number of nonregistered signals.