(Papers) ACET Paper June 2015 "CT4 Models"

Q.A1 Two trains 2301 and 2401 stop at Dadar and go to Churchgate. I am waiting at Dadar. 2301 is a “fast” train and has fewer stops and hence takes lesser time. Both the trains arrive at Dadar in accordance with independent Poisson processes with parameters 5 per hour and 1 per hour respectively.

i) Calculate the probability that
a) Exactly six 2301’s and no 2401 to arrive in the next one hour
b) At least 3 trains arrive in the next half an hour
c) I have to wait for more than 15 minutes for a train, if I have just missed one
d) I see exactly one 2301 pass while I am waiting for a 2401

ii) Suppose I need to get to a stop that only 2301 services and not 2401. If I forget this fact and just take the first train that arrives, what is the probability that I catch the wrong train.

iii) If half the 2401’s and a third of the 2301’s are manufactured at Chittaranjan, independently of anything else, how long do I have to wait for a Chittaranjan manufactured train, if one hasn’t arrived for over an hour.

Q.A2 For a discrete time stochastic process Xn , define the terms

Stationary
Weakly stationary
Increment
Markov property
Martingale

Q.A3 A fair coin is tossed repeatedly. Every time it lands heads, Aishwarya pays Vivek Re. 1 and every time it hands tails, Vivek pays Aishwarya Re. 1. Vivek is assumed to have access to infinite resources (essentially assumed to be very rich). Let Aishwarya’s available funds at time n be denoted by Sn. Assume that she starts with Rs. k and that if she goes broke she will have to stop playing but if she reaches Rs. K she will quit while ahead; where 0<k<K.

a) What process that Sn follow? What is the initial condition? What are the boundary conditions?
b) Show that Sn is a Martingale.
c) Define the term stopping time. State the Optimal Stopping Theorem.
d) Find the probability that Aishwarya is ultimately ruined.
e) If Aishwarya gets greedy and does not quit while ahead, what is her ultimate probability of ruin.
f) Why have we assumed that Vivek is very rich?