5.8 An electronic signboard in a sub- way station displays W, the time (in whole

Question

5.8 An electronic signboard in a sub-
way station displays W, the time (in
whole minutes) since the last train ar-
rived. Note that the signboard only
displays integers: if the last train ar-
rived t = 3:02 minutes ago, the sign-
board displays W = btc = 3. The in-
stant a train arrives, the signboard re-
sets to W = 0. Curiously, the trains
follow a synchronized schedule: interar-
rival times are independent and are ex-
actly K minutes long such that K is a
discrete uniform (2; 5) random variable.
That is, PK(k) = 1=4 for k = 2; 3; 4; 5.
If you arrive at a random time, what
is the probability 3 that the signboard
says W = 3?

5.6. In In the following problem, time l passes in slots indexed by the integer r 1,2, In each slot t, a visitor passes fo by a dog Charm with probability a, in- ra dependent of any other slot. If Charm is th I MCs PROBLEM 5.8. An electronic signboard in a sub- cle way station displays W, time pe whole minutes) since the last train ar po that the signboard only fou displays integers: if the last train ar the rived t 3.02 minutes ago, the sign ate board displays W t 3. The in the stant a train arrives, the signboard re- the sets to W 0. Curiously, the trains vidi follow a synchronized schedule: rival times are independent and are ex (a) actly K minutes long such that K is a discrete uniform (2, 5) random variable That is, PK(k) 1/4 for k 2, 3, 4, 5. If you arrive at a random time, what is the probability that the signboard says W 3? 5.9 In this problem, we extend the ran dom walk of Problem 5.3 to have posi- tions 0, 1. li. In particular, the state 5.11 transitions are ing a rolls a Given next t

Explanation / Answer

W = 3 when the interarrival time is between 3 and 4 minutes

f(x) follows uniform (2,5)

P(3<x<4) = P(x<4) - P(x<3)

=> F(4) - F(3)

=>2/3 - 1/3

=> 1/3

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