Stochastic Processes
42 questions across 3 topics. Also examined in CNET, ISS, GATE ST.
Markov Chains: Classification of States, Stationary Distribution
- 119Stationary distributions of two-state chainsMSET June 2025Markov Chains: Classification of States, Stationary DistributionHard
- 120First-passage probabilities and closed classesMSET June 2025Markov Chains: Classification of States, Stationary DistributionHard
- 121Transience via the return seriesMSET June 2025Markov Chains: Classification of States, Stationary DistributionHard
- 122First-return decomposition for a periodic chainMSET June 2025Markov Chains: Classification of States, Stationary DistributionHard
- 119Markov chains — the two equivalent faces of null persistenceMSET April 2024Markov Chains: Classification of States, Stationary DistributionHard
- 120Doubly stochastic matrices — solving for a missing entryMSET April 2024Markov Chains: Classification of States, Stationary DistributionHard
- 121Markov chain simulation — inverting the cumulative rowMSET April 2024Markov Chains: Classification of States, Stationary DistributionHard
- 122Markov chains — a self-contained pair of statesMSET April 2024Markov Chains: Classification of States, Stationary DistributionHard
- 119Mean recurrence time versus first-return probabilityMSET March 2023Markov Chains: Classification of States, Stationary DistributionMedium
- 120First-visit probabilities into a closed class — when they're forced to be 1, and when they're notMSET March 2023Markov Chains: Classification of States, Stationary DistributionHard
- 121First-return probability is bounded by return probability; recurrence forces divergenceMSET March 2023Markov Chains: Classification of States, Stationary DistributionMedium
- 122Non-null persistence (not transience, null persistence, or mere ergodicity) is the exact requirementMSET March 2023Markov Chains: Classification of States, Stationary DistributionMedium
- 124A pure death process — one absorbing state, everything else transientMSET March 2023Markov Chains: Classification of States, Stationary DistributionMedium
- 148Unbiasedness for a symmetric uniform location family — checking each candidateMSET September 2021Markov Chains: Classification of States, Stationary DistributionMedium
- 176Two disjoint closed classes give infinitely many stationary distributionsMSET September 2021Markov Chains: Classification of States, Stationary DistributionHard
- 179Positive recurrence guarantees a stationary distribution — but not, without irreducibility, a unique oneMSET September 2021Markov Chains: Classification of States, Stationary DistributionHard
- 176Marginalizing a mixed initial distribution through one step of the transition matrixMSET December 2020Markov Chains: Classification of States, Stationary DistributionEasy
- 179A guaranteed but finite-mean return to 0 makes every state positive recurrentMSET December 2020Markov Chains: Classification of States, Stationary DistributionHard
- 176An irreducible chain of period 3: a unique stationary law, but no limitMSET June 2019Markov Chains: Classification of States, Stationary DistributionMedium
- 178A symmetric walk reflected at 0 is null recurrentMSET June 2019Markov Chains: Classification of States, Stationary DistributionMedium
- 135States 0, 1, 2 all communicate: the chain is irreducibleMSET January 2018 Paper IIIMarkov Chains: Classification of States, Stationary DistributionEasy
- 136The two-state chain's limit is (beta,alpha)/(alpha+beta)MSET January 2018 Paper IIIMarkov Chains: Classification of States, Stationary DistributionEasy
- 133An inessential state is always transient — statement 3 is true tooMSET April 2017 Paper IIIMarkov Chains: Classification of States, Stationary DistributionMedium
- 134One-step transition probability p12=tfrac14MSET April 2017 Paper IIIMarkov Chains: Classification of States, Stationary DistributionEasy
- 135A doubly stochastic two-state chain has the uniform stationary lawMSET April 2017 Paper IIIMarkov Chains: Classification of States, Stationary DistributionEasy
Poisson and Birth-and-Death Processes
- 123Poisson process — waiting for the first arrivalMSET June 2025Poisson and Birth-and-Death ProcessesHard
- 123Poisson process — probability of a silent intervalMSET April 2024Poisson and Birth-and-Death ProcessesEasy
- 124Pure birth process with constant rate — recognising a Poisson processMSET April 2024Poisson and Birth-and-Death ProcessesHard
- 177Covariance between a Poisson process value and an independent random timeMSET September 2021Poisson and Birth-and-Death ProcessesMedium
- 178Extinction probabilities for three branching processesMSET September 2021Poisson and Birth-and-Death ProcessesHard
- 177Conditional on the Poisson count, event times are order statistics of a uniform sampleMSET December 2020Poisson and Birth-and-Death ProcessesEasy
- 178Summing a geometric series of branching-process generation meansMSET December 2020Poisson and Birth-and-Death ProcessesEasy
- 177min(s,t)/sqrtst is the Poisson process correlation; the covariance is lambdamin(s,t)MSET June 2019Poisson and Birth-and-Death ProcessesEasy
- 179A finite expected total progeny forces m<1 and certain extinctionMSET June 2019Poisson and Birth-and-Death ProcessesMedium
- 137An incomplete birth-process event with no correct optionMSET January 2018 Paper IIIPoisson and Birth-and-Death ProcessesEasy
- 138Poisson with mean 6: P(N4)=1-115e^-6MSET April 2017 Paper IIIPoisson and Birth-and-Death ProcessesEasy
Martingales
- 139Building martingales from mean-zero, unit-variance incrementsMSET September 2021MartingalesHard
- 139A recursively-defined process can also be a martingale, but adding a deterministic drift breaks itMSET December 2020MartingalesHard
- 139A square-integrable martingale's second moment is non-decreasing, not constantMSET June 2019MartingalesMedium
- 83A martingale remains one for its own (smaller) filtrationMSET January 2018 Paper IIIMartingalesEasy
- 81A product of independent mean-one factors is a martingaleMSET April 2017 Paper IIIMartingalesEasy
- 82Doob's martingale E(Zmid Fn)MSET April 2017 Paper IIIMartingalesEasy