A message requires N time units to be transmitted, where N is a geometric random variable

with PMF pN(i) = (1 ? a)a^(i?1) for i = 1, 2, 3, .... A single new message arrives during a time

unit with probability p, and no messages arrive with probability 1?p. Let K be the number

of new messages that arrive during the transmission of a single message.

(a) Find E[K] and VAR[K] using conditional expectation.

(b) Find the PMF of K.

(c) Find the conditional PMF of N given K = k.

(d) Find the value of n that maximizes P[N = n|X = k].

A message requires N time units to be transmitted, where N is a geometric random variable with PMF pN(z') : (1 (val1 for i : 1, 2, 3, A single new message arrives during a time unit with probability p, and no messages arrive with probability 1 p. Let K be the number of new messages that arrive during the transmission of a single message. (a) Find E[K] and VAR[K] using conditional expectation. (b) Find the PMF of K. (Hint: (1 gram) : :ka (gmk.) (c) Find the conditional PMF of N given K : k. (d) Find the value of n that maximizes P[N : n|X : k]