Please help with all sections of a,b,c, and d: Thanks in Advance:)

PROBLEM 3 In a binary optical communication system, the receiver has a detector that counts the number of photoelectrons ejected by the incident light on a photo cell during a given interval (0,T). When no light signal has been transmitted towards the detector, {evet Ho }, the probability mass function of the electron count, K, is given by P [K|Ho] = (1 - vo)Vo, k = 0,1,2,3 .. When a signal has been transmitted, P [K| H, ] = (1 - v,)va, k = 0,1,2,3 .. The prior probabilities of a signal has been transmitted and no signal transmitted are, P[H, ] = p and P[Ho] = 1 - p. a. Using the law of total probability, write an expression for the probability that the number of photoelectrons is equal to m, P[K = m]. b. Given the number of electrons K = m, what is the probability that a signal has been transmitted, P[H, |K = m]. What is P [Ho|K = m] c. If the maximum a posteriori probability, MAP decision rule is used to decide whether or not a signal H1 has been transmitted P[H, |K = m] 2 P[H,|K = m], show that the decision rule will be in the HoH1 form of : m 2 no, which compares the electron count to a threshold count no. If the count is larger Ho than or equal to the threshold no, the decision is made in favor or Hj, otherwise the decision is made in favor or Ho d. For P[H.] = P[H,] = = and v. = 0.1, v, = 0.9, find the threshold count n, and the probability of error. Note that an error will occur if a signal has been transmitted and K