In the message transmission system in Problem 5.3.2, the solution to Problem 5.5.2 is a formula for the PMF of J, the number of transmissions of individual messages. For p = 0.8, find the expected value vector E[J], the correlation matrix R_J, and the covariance matrix C_J.

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

(a) Find the PMF of K.

(b) For each j ∈ {1, 2,..., n - 1}, find the marginal PMF PK₁,K₂,...,K j(k₁, k₂,..., k_j)

(c) For each i ∈ {1, 2,..., n}, find the marginal PMF PKi(ki).