China national postgraduate entrance exam - Statistics (0714) and Applied Statistics master's (432) · China · Ministry of Education / China Higher Education Student Information (CHSI) with universities
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 China national postgraduate entrance exam spread across the units of Stochastic Processes. Difficulty: 23 easy, 40 medium, 13 hard.
| Classification of states and periodicity | 10 | |
| Poisson process and its properties | 9 | |
| Markov chains: transition matrices and Chapman-Kolmogorov equations | 8 | |
| Limiting and stationary distributions | 8 | |
| Pure birth, pure death and birth-death processes | 8 | |
| Queueing models M/M/1 and M/M/c | 8 | |
| Branching processes and extinction probability | 7 | |
| Brownian motion and its basic properties | 6 | |
| Definition and classification of stochastic processes | 5 | |
| Renewal processes (introduction) | 4 | |
| Markov-chain models of rainfall and crop sequences | 3 |
Choose an answer, then open the working. Each explanation says why the right answer is right and why the others are not.
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
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
Farmers arrive at a soil-testing lab as a Poisson process with rate 3 per hour. What is the probability of no arrivals in 30 minutes?
D. 0.223
N(0.5) ~ Poisson(1.5): P(0) = e^(−1.5) = 0.223. e^(−3) uses a full hour.
Further reading: Karlin and Taylor, A First Course in Stochastic Processes; Ross, Introduction to Probability Models; Medhi, Stochastic Processes
A biocontrol agent has per-individual birth rate 0.3 and death rate 0.2 per day (linear birth-death). What is the probability of eventual extinction if 5 individuals are released?
D. 0.132
For λ > μ, extinction from i individuals has probability (μ/λ)ⁱ = (2/3)⁵ = 0.132. 0.667 is the risk from a single individual.
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
A single introduction has extinction probability 0.5. What is the extinction probability with 10 independent founders?
A. 0.0010
Independent lines must all die out: qᵏ = 0.5¹⁰ = 0.0010.
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
In a wet/dry chain with P(wet | wet) = 0.6, what is the mean length of a wet spell?
B. 2.5 days
Wet-spell length is geometric with mean 1/(1 − p₁₁) = 1/0.4 = 2.5 days. 1/0.6 uses the persistence probability itself.
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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