I am a non-student working through the first edition of Yates and Goodman's text, Probability and Stochastic Processes. On page 115, question 3.6.8 goes like this:
Suppose you arrive at a bus stop at time 0 and that at the end of each minute, with probability $p$ a bus arrives or with probability $1-p,$ no bus arrives. Whenever a bus arrives, you board that bus with probability $q$ and depart. Let $T$ equal the number of minutes you stand at a bus stop. Let $N$ be the number of buses that arrive while you wait at the bus stop.
Now, there are four parts to this problem; the second part is where I am struggling:
Find $P_{N,T}(n,t).$
I understand that $P_{N,T}(n,t) = P_{N|T}(n|t) \cdot P_T(t).$ However, I'm unsure how to proceed because I'm unsure of which discrete random distribution applies here. At first I thought this was a case where using a binomial random variable would suffice; however, my textbook also brings up the "Pascal random variable," which as I understand it, features a random variable that is the number of trials up to and including the $k^{th}$ success.
Is it fair to say that time $T$ (in this problem) is a Pascal random variable? Could $N$ be one as well?
Thank you in advance.