A wireless data terminal has three messages waiting for

Question

From the joint PMF P_K(k) in Problem 5.3.2, find the marginal PMFs

Problem 5.3.2

A wireless data terminal has three messages waiting for transmission. After sending a message, it expects an acknowledgement from the receiver. When it receives the acknowledgement, it transmits the next message. If the acknowledgement does not arrive, it sends the message again. The probability of successful transmission of a message is p independent of other transmissions. Let K = [K₁ K₂ K₃] be the 3-dimensional random vector in which K_i is the total number of transmissions when message i is received successfully. (K₃ is the total number of transmissions used to send all three messages.) Show that