Indian Statistical Institute admission tests (BStat, MStat, JRF in Statistics) · India · Indian Statistical Institute
Original practice questions written to the published syllabus · not past papersRandom processes in time: Markov chains and their classification and limits, Poisson and birth-death processes, branching processes and Brownian motion, applied to population growth, epidemics, queues and weather sequences such as wet- and dry-day chains.
In the exams: GATE ST 'Stochastic Processes' (heavy); UPSC ISS Statistics I (present); ISI admission (present); CSIR-NET Unit 4 (present); light in AICE-JRF Ag Stat
How our questions for Indian Statistical Institute admission tests spread across the units of Stochastic Processes. Difficulty: 15 easy, 13 medium, 2 hard.
| Definition and classification of stochastic processes | 3 | |
| Markov chains: transition matrices and Chapman-Kolmogorov equations | 3 | |
| Classification of states and periodicity | 3 | |
| Limiting and stationary distributions | 3 | |
| Pure birth, pure death and birth-death processes | 3 | |
| Branching processes and extinction probability | 3 | |
| Brownian motion and its basic properties | 3 | |
| Renewal processes (introduction) | 3 | |
| Poisson process and its properties | 2 | |
| Queueing models M/M/1 and M/M/c | 2 | |
| Markov-chain models of rainfall and crop sequences | 2 |
Choose an answer, then open the working. Each explanation says why the right answer is right and why the others are not.
What is a sample path of a stochastic process?
C. One realisation over time
Fixing the random outcome ω gives a function of t, the trajectory actually observed, e.g. one season's daily rainfall sequence.
Further reading: Karlin and Taylor, A First Course in Stochastic Processes; Ross, Introduction to Probability Models; Medhi, Stochastic Processes
When is state i of a Markov chain recurrent?
A. Return to i is certain
Recurrence means fᵢᵢ = 1, equivalently Σₙ pᵢᵢ⁽ⁿ⁾ = ∞. pᵢᵢ > 0 concerns aperiodicity, not recurrence.
Further reading: Karlin and Taylor, A First Course in Stochastic Processes; Ross, Introduction to Probability Models; Medhi, Stochastic Processes
In a birth-death process in state n, how long does the process stay before the next transition?
A. Exponential with rate λₙ + μₙ
Competing exponential clocks for birth and death give an exponential holding time with the summed rate; the move is a birth with probability λₙ/(λₙ + μₙ).
Further reading: Karlin and Taylor, A First Course in Stochastic Processes; Ross, Introduction to Probability Models; Medhi, Stochastic Processes
An invasive pest has offspring mean slightly above 1. Why can early eradication still succeed often?
D. Chance extinction of few founders
For m just above 1 the extinction probability q is close to 1, so a few founders often fail to establish; detection and removal raise the chance further.
Further reading: Karlin and Taylor, A First Course in Stochastic Processes; Ross, Introduction to Probability Models; Medhi, Stochastic Processes
In Kendall's notation, what does M/M/2 denote?
B. Poisson arrivals, exponential service, two servers
A/B/c: arrival process, service distribution, number of servers; M stands for Markovian (Poisson/exponential).
Further reading: Karlin and Taylor, A First Course in Stochastic Processes; Ross, Introduction to Probability Models; Medhi, Stochastic Processes
How does a renewal process differ from a Poisson process?
C. Inter-event times need not be exponential
A renewal process keeps iid inter-event times but drops the exponential assumption; the Poisson process is the exponential special case.
Further reading: Karlin and Taylor, A First Course in Stochastic Processes; Ross, Introduction to Probability Models; Medhi, Stochastic Processes
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