CSIR-UGC NET - Mathematical Sciences (statistics option) · India · National Testing Agency for CSIR
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 CSIR-UGC NET spread across the units of Stochastic Processes. Difficulty: 0 easy, 16 medium, 22 hard.
| Definition and classification of stochastic processes | 5 | |
| Classification of states and periodicity | 5 | |
| Brownian motion and its basic properties | 5 | |
| Renewal processes (introduction) | 5 | |
| Limiting and stationary distributions | 4 | |
| Branching processes and extinction probability | 4 | |
| Pure birth, pure death and birth-death processes | 3 | |
| Markov chains: transition matrices and Chapman-Kolmogorov equations | 2 | |
| Poisson process and its properties | 2 | |
| Markov-chain models of rainfall and crop sequences | 2 | |
| Queueing models M/M/1 and M/M/c | 1 |
Choose an answer, then open the working. Each explanation says why the right answer is right and why the others are not.
Which statement about independent increments and the Markov property is correct?
D. Independent increments imply the Markov property, but not conversely
If increments are independent, the future increment ignores the past, so the process is Markov. A wet/dry rainfall chain is Markov but its 'increments' are not independent.
Further reading: Karlin and Taylor, A First Course in Stochastic Processes; Ross, Introduction to Probability Models; Medhi, Stochastic Processes
How is a process whose next value depends on its last two values made into a Markov chain?
B. Enlarge the state
Enlarging the state to the last two values gives a first-order Markov chain on pairs; this is how second-order rainfall chains are handled.
Further reading: Karlin and Taylor, A First Course in Stochastic Processes; Ross, Introduction to Probability Models; Medhi, Stochastic Processes
For a two-state chain with p₀₁ = 0.3 and p₁₀ = 0.4, at what geometric rate do the rows of Pⁿ approach their limit?
A. 0.3ⁿ
The second eigenvalue of P is 1 − a − b = 0.3, and Pⁿ = limit + (1 − a − b)ⁿ × (fixed matrix). 0.7 = a + b and 0.4 = p₁₀ are not eigenvalues of P.
Further reading: Karlin and Taylor, A First Course in Stochastic Processes; Ross, Introduction to Probability Models; Medhi, Stochastic Processes
Which criterion characterises a recurrent state i?
A. Σₙ pᵢᵢ⁽ⁿ⁾ = ∞
The expected number of returns, Σₙ pᵢᵢ⁽ⁿ⁾, is infinite for recurrent and finite for transient states. pᵢᵢ = 1 defines an absorbing state, a special recurrent case.
Further reading: Karlin and Taylor, A First Course in Stochastic Processes; Ross, Introduction to Probability Models; Medhi, Stochastic Processes
According to Pólya's theorem, in which dimensions is the symmetric simple random walk recurrent?
D. One and two dimensions only
The return probability is 1 in dimensions 1 and 2 and below 1 in dimensions 3 and higher.
Further reading: Karlin and Taylor, A First Course in Stochastic Processes; Ross, Introduction to Probability Models; Medhi, Stochastic Processes
A random walk on 0, ..., 6 steps up with probability 0.6 and down with 0.4, starting at 3 and absorbed at 0 and 6. What is the probability of reaching 6 first?
C. 0.7714
[1 − (q/p)ⁱ]/[1 − (q/p)ᴺ] with q/p = 2/3: [1 − (2/3)³]/[1 − (2/3)⁶] = 0.7714. 0.5 would hold only for p = 1/2.
Further reading: Karlin and Taylor, A First Course in Stochastic Processes; Ross, Introduction to Probability Models; Medhi, Stochastic Processes
A periodic irreducible positive-recurrent chain has stationary distribution π. Which statement is correct?
C. Long-run time fractions equal π, though pᵢⱼ⁽ⁿ⁾ itself does not converge
Periodicity prevents convergence of n-step probabilities, but averages over time still converge to π (the ergodic theorem needs no aperiodicity).
Further reading: Karlin and Taylor, A First Course in Stochastic Processes; Ross, Introduction to Probability Models; Medhi, Stochastic Processes
Insect emergence follows a non-homogeneous Poisson process with rate λ(t) = 2t per day for 0 ≤ t ≤ 3. What is the expected number of emergences in the first 3 days?
B. 9
Mean = ∫₀³ 2t dt = t² evaluated at 3 = 9. Using λ(3) × 3 = 18 overstates it; 6 is λ(3).
Further reading: Karlin and Taylor, A First Course in Stochastic Processes; Ross, Introduction to Probability Models; Medhi, Stochastic Processes
What does the renewal-reward theorem give?
B. Long-run reward per unit time = E(reward per cycle)/E(cycle length)
With iid cycles, total reward up to t divided by t converges almost surely to E(R)/E(X).
Further reading: Karlin and Taylor, A First Course in Stochastic Processes; Ross, Introduction to Probability Models; Medhi, Stochastic Processes
Field surveys estimate infectious periods by measuring animals that are infected on the survey day. What bias results?
D. Long periods are over-sampled
Units sampled at a fixed time are more likely to be in long intervals (the inspection paradox), so naive averages overestimate the mean duration.
Further reading: Karlin and Taylor, A First Course in Stochastic Processes; Ross, Introduction to Probability Models; Medhi, Stochastic Processes
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