A node in a wireless network experiences one among three possibilities in any wireless timeslot. In

any slot, the node either idles $($outcome I$)$ or transmits successfully $($T$)$ or transmits unsuccessfully U$,$ independently

of outcomes in any other slot. Probability that a node idles in a slot is p$,$ that it transmits successfully is q$,$ and

that it transmits unsuccessfully is r$.$ Answer the following questions.

$($a$)$ What is the probability that the node transmits successfully in the first slot, has an unsuccessful transmission

in the second slot, and a successful transmission in the third?

$($b$)$ You are interested in characterizing the random number of slots up to and including the slot in which the node

transmits successfully for the first time. Derive the distribution $($PMF$)$ of this number.

$($c$)$ Derive the probability that the first successful transmission slot occurs earlier than the first unsuccessful

transmission slot. $[$Hint: Think of a suitable event space.$]$

$($d$)$ Suppose that the first successful transmission slot is observed by the node before the first unsuccessful transmis$-$

sion slot. Derive the PMF of the random number of slots including and up to the first successful transmission.

$[$Hint: The result you derived in $($c$)$ may be useful$].$

$($e$)$ As in $($d$),$ suppose that the first successful transmission slot is observed by the node before the first unsuccessful

transmission slot. Derive the expected value of the number of slots up to and including the first unsuccessful

transmission slot? $[$Hint: Think sum of expected values.$]$