At each time, the transmitter sends out either a '1' with probability p, or a '0' with probability 1-p. Given that the transmitted bit is '1', the receiver successfully receives the '1' with probability of or it receives an incorrect message of 'O' with probability 1 - 01. Similarly, given that the transmitted bit is 'O', the receiver either receives a '0' with probability oo, or a '1' with probability 1 -oo, respectively. (a) Suppose the number of transmissions within a given observation window T has a Poisson PMF with average rate A. Define a random variable X to be the number of 1's transmitted in that time interval. Show that X has a Poisson PMF with average rate Ap. (Hint: Define V = total transmitted bits in the given interval. Try to use the total probability theorem P(X = k) = En_, P(X = k(V =k +n) . P(V = k +n)) (b) By using the setting in (a), we suppose that the number of transmissions within a given observation window I has a Poisson PMF with average rate A. Define a random variable Y to be the total number of 1's received in that time interval. What is the PMF of Y