September 2021
190 questions, booklet SEP-30221.
Section I · General (compulsory) · Q1–10
Practise this section →- 1Positive convergent sequences — what survives to every subsequenceAnalysisSequences and SeriesHard
- 2A monotone function landing in mathbf Q must be constant somewhereAnalysisMonotonic Functions, Bounded Variation, Lebesgue Measure & IntegralHard
- 3A universal inequality between x+y and zComplex AnalysisAlgebra of Complex Numbers, Power Series, Transcendental FunctionsEasy
- 4Complex cosine — unbounded, not injective, yet surjectiveComplex AnalysisAlgebra of Complex Numbers, Power Series, Transcendental FunctionsMedium
- 5Dimension of the space of polynomials of bounded degreeLinear AlgebraVector Spaces, Subspaces, Linear Dependence, Basis, DimensionEasy
- 6Injective linear maps — what they preserve, and what they don't forceLinear AlgebraVector Spaces, Subspaces, Linear Dependence, Basis, DimensionHard
- 7A direct conditional-probability computationProbabilitySample Space, Discrete Probability, Independent Events, Bayes TheoremEasy
- 8A basic hypergeometric probabilityDistribution TheoryStandard Discrete and Continuous Univariate DistributionsEasy
- 9Recognising convex sets — a line, a wedge, and an epigraphAnalysisElementary Set Theory, Real Number System, Supremum, InfimumMedium
- 10Convexity facts in linear programmingOperations ResearchLinear Programming Problem, Simplex Method, DualityMedium
Section II · Mathematics · Q11–100
Practise this section →- 11Eventually-zero sequences are exactly the countable onesAnalysisSequences and SeriesHard
- 12Bernoulli's inequality — pinning down the valid range of xAnalysisContinuity, Uniform Continuity, Differentiability, Mean Value TheoremMedium
- 13Counting the crossings of cos x and e^-xAnalysisElementary Set Theory, Real Number System, Supremum, InfimumMedium
- 14Bounded variation implies Riemann integrable — and nothing strongerAnalysisRiemann Sums and Riemann Integral, Improper IntegralsHard
- 15Non-degenerate open and connected subsets of mathbf R are always uncountableAnalysisElementary Set Theory, Real Number System, Supremum, InfimumMedium
- 16The standard Möbius map from the unit disc to the right half-planeComplex AnalysisConformal Mappings, Mobius TransformationsEasy
- 17The zero set of a nonzero entire function is always closed (and discrete, and countable)AnalysisElementary Set Theory, Real Number System, Supremum, InfimumMedium
- 18Harmonic combinations of u,v — why u^2 fails but uv survivesComplex AnalysisAnalytic Functions, Cauchy-Riemann EquationsMedium
- 19Contour integrals around concentric circles agree, by deformationComplex AnalysisTaylor Series, Laurent Series, Calculus of ResiduesMedium
- 20A double pole with vanishing residue — none of the printed options surviveAnalysisMonotonic Functions, Bounded Variation, Lebesgue Measure & IntegralHard
- 21Equal subgroups characterised via a shared cosetAlgebraGroups, Subgroups, Sylow Theorems, Cayley's TheoremHard
- 22An ideal generated by a unit is the whole ringAlgebraRings, Ideals, UFD, PID, Euclidean DomainMedium
- 23Counting elements of each order in S3AlgebraGroups, Subgroups, Sylow Theorems, Cayley's TheoremMedium
- 24Counting order-10 elements in a cyclic group of order 20AlgebraGroups, Subgroups, Sylow Theorems, Cayley's TheoremEasy
- 25A domain that is not a field — the polynomial ring over mathbf RAlgebraRings, Ideals, UFD, PID, Euclidean DomainEasy
- 26Powers of a unipotent Jordan blockLinear AlgebraMatrix Representation, Change of Basis, Canonical and Jordan FormsEasy
- 27Dimension of the skew-symmetric matricesLinear AlgebraVector Spaces, Subspaces, Linear Dependence, Basis, DimensionEasy
- 28A real 3x3 matrix must have a real eigenvalue — ruling out a purely-complex minimal polynomialLinear AlgebraAlgebra of Matrices, Rank, Determinant, Linear EquationsHard
- 29A block matrix with a genuinely complex 2times2 blockLinear AlgebraAlgebra of Matrices, Rank, Determinant, Linear EquationsMedium
- 30Orthogonal complement of a subspace defined by two linear equationsLinear AlgebraInner Product Spaces, Orthonormal BasisMedium
- 31Eliminating two constants from z=ae^bxsin(by) — Laplace's equation appearsPartial Differential EquationsLagrange and Charpit Methods, Cauchy Problem for First Order PDEsMedium
- 32A repeated-characteristic-root PDEPartial Differential EquationsSeparation of Variables (Laplace, Heat and Wave Equations)Medium
- 33Abel's formula — no y' term means a constant WronskianOrdinary Differential EquationsGeneral Theory of Linear ODEs, Variation of Parameters, Sturm-Liouville, Green's FunctionMedium
- 34Solving xy'=2y and tracking a specific solution curveOrdinary Differential EquationsGeneral Theory of Linear ODEs, Variation of Parameters, Sturm-Liouville, Green's FunctionEasy
- 35Eliminating both constants from y=ae^bxOrdinary Differential EquationsGeneral Theory of Linear ODEs, Variation of Parameters, Sturm-Liouville, Green's FunctionEasy
- 36Assignment problem — minimising total programmer timeOperations ResearchTransportation and Assignment ProblemsMedium
- 37A two-person zero-sum game with a saddle pointOperations ResearchGame TheoryEasy
- 38Contradictory constraints make an LP infeasibleOperations ResearchLinear Programming Problem, Simplex Method, DualityEasy
- 39Constructing the dual of a three-constraint LPPOperations ResearchLinear Programming Problem, Simplex Method, DualityMedium
- 40The standard basic-variable count for a transportation tableOperations ResearchTransportation and Assignment ProblemsEasy
- 41Integrating an indicator that is 1 almost everywhereAnalysisMonotonic Functions, Bounded Variation, Lebesgue Measure & IntegralEasy
- 42Implicit function theorem — checking each partial derivative at the originAnalysisFunctions of Several Variables, Inverse & Implicit Function TheoremsMedium
- 43The complex-squaring map is an open map away from its critical pointComplex AnalysisAlgebra of Complex Numbers, Power Series, Transcendental FunctionsMedium
- 44Picard's theorem — what a non-constant entire function's range can look likeComplex AnalysisLiouville's Theorem, Maximum Modulus Principle, Open Mapping TheoremMedium
- 45Maxima and minima of two composed complex functionsAnalysisFunctions of Several Variables, Inverse & Implicit Function TheoremsHard
- 46Consequences of sub-linear growth for a non-constant entire functionAnalysisSequences and SeriesMedium
- 47Endomorphisms of an algebraic extension are automatically bijectiveAlgebraFields, Finite Fields, Field Extensions, Galois TheoryHard
- 48X^16+1 has no real roots — hence no linear factorsAlgebraPolynomial Rings and Irreducibility CriteriaEasy
- 49Counting irreducible quadratic factors of X^8+1Complex AnalysisAlgebra of Complex Numbers, Power Series, Transcendental FunctionsEasy
- 50Distance between disjoint sets — compactness (not just closedness) is what's neededAnalysisContinuity, Uniform Continuity, Differentiability, Mean Value TheoremHard
- 51Two classical oscillating functions — neither is of bounded variationAnalysisMonotonic Functions, Bounded Variation, Lebesgue Measure & IntegralHard
- 52Continuous surjections preserve densityAnalysisMetric Spaces, Compactness, Connectedness, Normed Linear SpacesMedium
- 53The correct Sylow-counting formulaAlgebraGroups, Subgroups, Sylow Theorems, Cayley's TheoremMedium
- 54Counting order-3 elements in mathbf Z9timesmathbf Z3AlgebraGroups, Subgroups, Sylow Theorems, Cayley's TheoremEasy
- 55Comparing degree-8 and degree-2 field extensionsAlgebraFields, Finite Fields, Field Extensions, Galois TheoryMedium
- 56A bijective bounded linear map transported to a pair of isometric renormingsAnalysisMetric Spaces, Compactness, Connectedness, Normed Linear SpacesHard
- 57Finitely-supported subspaces of ell^2 never algebraically sum to all of ell^2AnalysisSequences and SeriesHard
- 58A multiplication operator is always normal, but rarely self-adjoint or unitaryAnalysisMetric Spaces, Compactness, Connectedness, Normed Linear SpacesHard
- 59Transporting continuity across coarser/finer topologiesAnalysisContinuity, Uniform Continuity, Differentiability, Mean Value TheoremHard
- 60A chain of overlapping unit circles — connected, but unboundedTopologySeparation Axioms, Connectedness and CompactnessMedium
- 61Finite products preserve both first countability and separabilityAnalysisElementary Set Theory, Real Number System, Supremum, InfimumMedium
- 62The subgroup lattice of a cyclic group is a product of chainsAlgebraGroups, Subgroups, Sylow Theorems, Cayley's TheoremMedium
- 63Forbidding an induced 2-edge subgraph forces G to be completeGraph TheoryGraphs, Trees, ConnectivityMedium
- 64A coefficient forced to vanish by a minimum-degree argumentAlgebraPermutations, Combinations, Pigeon-Hole Principle, Inclusion-ExclusionEasy
- 65Neumann problems for Laplace's equation determine u only up to a constantPartial Differential EquationsSeparation of Variables (Laplace, Heat and Wave Equations)Medium
- 66Classifying a variable-coefficient second-order PDE — the maximal hyperbolic regionPartial Differential EquationsSeparation of Variables (Laplace, Heat and Wave Equations)Easy
- 67Classifying a first-order PDE with z-dependent coefficientsAnalysisFunctions of Several Variables, Inverse & Implicit Function TheoremsEasy
- 68Binary representation — sums of distinct powers of 2AlgebraPermutations, Combinations, Pigeon-Hole Principle, Inclusion-ExclusionEasy
- 69Solving a system of three congruences by direct verificationAlgebraFundamental Theorem of Arithmetic, Congruences, Chinese Remainder TheoremEasy
- 70Using Wilson's theorem to evaluate (50!)^2bmod101AlgebraFundamental Theorem of Arithmetic, Congruences, Chinese Remainder TheoremEasy
- 71Degrees of freedom — the correct constraint-counting formulaClassical MechanicsGeneralized Coordinates, Lagrange's EquationsMedium
- 72Canonical momenta from a Lagrangian with a velocity-coupling termClassical MechanicsHamilton's Canonical Equations, Hamilton's PrincipleEasy
- 73Legendre transform back from a given HamiltonianClassical MechanicsGeneralized Coordinates, Lagrange's EquationsMedium
- 74Finite rotations don't commute; infinitesimal ones doClassical MechanicsMotion of Rigid Bodies, Euler's Dynamical Equations, Small OscillationsMedium
- 75Checking the 2D continuity equation against four candidate flowsFluid DynamicsPotential Flow, Streamlines, DoubletsEasy
- 76The linear-velocity-field incompressibility conditionFluid DynamicsPotential Flow, Streamlines, DoubletsEasy
- 77Velocity components from a velocity potential (with the -nablaphi convention)Fluid DynamicsPotential Flow, Streamlines, DoubletsMedium
- 78Method of images — a source outside a circular cylinderFluid DynamicsPotential Flow, Streamlines, DoubletsHard
- 79The shape operator: real eigenvalues, but not always symmetric as a raw matrixDifferential GeometryCurves and Surfaces, CurvatureHard
- 80A Frenet-frame combination that collapses to zeroDifferential GeometryCurves and Surfaces, CurvatureMedium
- 81Cylinders are developable — Gaussian curvature vanishes identicallyDifferential GeometryCurves and Surfaces, CurvatureEasy
- 82An isoperimetric-free extremal problem — when the boundary data becomes inconsistentCalculus of VariationsVariation of a Functional, Euler-Lagrange Equation, Conditions for ExtremaMedium
- 83First integral of Euler–Poisson when f omits yCalculus of VariationsVariation of a Functional, Euler-Lagrange Equation, Conditions for ExtremaHard
- 84Solving a boundary value problem from Euler–LagrangeCalculus of VariationsVariation of a Functional, Euler-Lagrange Equation, Conditions for ExtremaEasy
- 85The brachistochrone is a cycloidCalculus of VariationsVariation of a Functional, Euler-Lagrange Equation, Conditions for ExtremaEasy
- 86Eigenfunctions for distinct eigenvalues are never linearly dependentOrdinary Differential EquationsGeneral Theory of Linear ODEs, Variation of Parameters, Sturm-Liouville, Green's FunctionHard
- 87Converting a trivial IVP into a Volterra integral equationOrdinary Differential EquationsExistence & Uniqueness, Singular Solutions, Systems of First Order ODEsEasy
- 88A symmetric-kernel operator is self-adjoint, with real "diagonal" inner productsLinear Integral EquationsFredholm and Volterra Type, Separable Kernels, Resolvent KernelMedium
- 89Third divided difference of 1/xNumerical AnalysisFinite Differences, Lagrange/Hermite/Spline InterpolationEasy
- 90Newton's forward-difference interpolation, worked to a cubicNumerical AnalysisFinite Differences, Lagrange/Hermite/Spline InterpolationEasy
- 91RK2 (Heun's method) for a single stepOrdinary Differential EquationsExistence & Uniqueness, Singular Solutions, Systems of First Order ODEsEasy
- 92A causal exponentially-weighted function — Fourier transform equals a Laplace transformIntegral TransformsLaplace TransformMedium
- 93Fourier transform of a time-shifted functionIntegral TransformsFourier TransformEasy
- 94Recognising a Fourier cosine transform pairLinear Integral EquationsFredholm and Volterra Type, Separable Kernels, Resolvent KernelEasy
- 95Two-phase simplex — a lingering zero-valued artificial signals degeneracyOperations ResearchLinear Programming Problem, Simplex Method, DualityHard
- 96Whitney's inequality — the only consistent connectivity tripleGraph TheoryGraphs, Trees, ConnectivityEasy
- 97Petersen's theorem — even-regular graphs decompose into 2-factorsGraph TheoryGraphs, Trees, ConnectivityMedium
- 98Evaluating a signed measure on an interval straddling the sign changeAnalysisMonotonic Functions, Bounded Variation, Lebesgue Measure & IntegralEasy
- 99Absolute continuity of an integral-defined measureAnalysisMonotonic Functions, Bounded Variation, Lebesgue Measure & IntegralHard
- 100A clean sufficient condition for absolute continuityAnalysisContinuity, Uniform Continuity, Differentiability, Mean Value TheoremMedium
Section III · Statistics · Q101–190
Practise this section →- 101A bimodal-looking dataset — mean, median and mode from raw countsDescriptive StatisticsDescriptive Statistics, Exploratory Data AnalysisMedium
- 102Coefficient of variation — invariant under rescalingProbabilityRandom Variables and Distribution Functions, Expectation and MomentsEasy
- 103Normal tail probabilities — comparing the mean to a point above itDistribution TheoryStandard Discrete and Continuous Univariate DistributionsEasy
- 104Bounding P(A) from a conditional probabilityProbabilitySample Space, Discrete Probability, Independent Events, Bayes TheoremEasy
- 105Markov's inequality on the third absolute momentProbabilityRandom Variables and Distribution Functions, Expectation and MomentsEasy
- 106Reading a probability off a joint tableProbabilityRandom Variables and Distribution Functions, Expectation and MomentsMedium
- 107Spotting a genuine characteristic function among four candidatesProbabilityCharacteristic Functions, Probability InequalitiesHard
- 108Unpacking a product-form independence hypothesisProbabilitySample Space, Discrete Probability, Independent Events, Bayes TheoremHard
- 109Density of the sample range of iid exponentialsDistribution TheoryStandard Discrete and Continuous Univariate DistributionsMedium
- 110Recognising a distribution from its moment generating functionDistribution TheoryStandard Discrete and Continuous Univariate DistributionsEasy
- 111Exponential distribution — picking the one universally true claimDemography / Vital StatisticsLife Tables, Mortality RateHard
- 112Linear combinations of iid trivariate normal vectorsMultivariate AnalysisMultivariate Normal Distribution, Wishart Distribution, Quadratic FormsHard
- 113MLE of a normal mean under a non-negativity constraintStatistical InferenceMethods of Estimation, Properties of EstimatorsHard
- 114Confidence interval from a single exponential observationStatistical InferenceConfidence IntervalsEasy
- 115Power and size of a min-based test for a uniform location parameterStatistical InferenceTests of Hypotheses: MP and UMP Tests, Likelihood Ratio TestsMedium
- 116Naming the theorem behind finding sufficient statisticsStatistical InferenceMethods of Estimation, Properties of EstimatorsMedium
- 117Exponential family — a complete sufficient statistic need not be unbiased for theta itselfStatistical InferenceMethods of Estimation, Properties of EstimatorsHard
- 118The classical assumptions behind the two-sample t-testProbabilityRandom Variables and Distribution Functions, Expectation and MomentsMedium
- 119The correct reading of a confidence intervalStatistical InferenceConfidence IntervalsHard
- 120Partial correlation — what it shares with, and how it differs from, ordinary correlationMultivariate AnalysisPartial and Multiple Correlation CoefficientsHard
- 121Degrees of freedom for a chi-square goodness-of-fit test with estimated parameters and merged cellsStatistical InferenceAnalysis of Discrete Data and Chi-Square Test of Goodness of FitEasy
- 122A one-sided sign test p-value from a tiny sampleStatistical InferenceNonparametric Tests, Rank Correlation, Test for IndependenceMedium
- 123Group-replacement policy — comparing average cost per weekOperations ResearchReplacement ModelsHard
- 124A newsvendor problem with an unstated order quantityProbabilityRandom Variables and Distribution Functions, Expectation and MomentsHard
- 125M/M/1 — probability the server is idleOperations ResearchSteady-State Solutions of Markovian Queuing Models (M/M/1, M/M/C, M/G/1)Medium
- 126A mixed unequal-probability/SRSWOR design — the sample mean is biased, but an unbiased estimator still existsSampling TheorySimple Random, Stratified and Systematic SamplingHard
- 127Regression estimator versus sample mean — an asymptotic, not universal, comparisonSampling TheorySimple Random, Stratified and Systematic SamplingHard
- 128Recognising a randomized block design in disguiseDesign of ExperimentsCompletely Randomized, Randomized Block and Latin-Square DesignsHard
- 129The full assumption set behind the one-way ANOVA F-testLinear Models & RegressionAnalysis of Variance and Covariance, Fixed/Random/Mixed EffectsMedium
- 130Identifying the confounded effect from Yates-notation block contentsDesign of Experiments2^K Factorial Experiments: Confounding and ConstructionMedium
- 131Set operations on sigma-fields — intersections always survive, unions generally don'tAnalysisMonotonic Functions, Bounded Variation, Lebesgue Measure & IntegralHard
- 132Measurability between two explicit finite sigma-fieldsProbabilityRandom Variables and Distribution Functions, Expectation and MomentsHard
- 133A Gaussian-weighted measure — absolutely continuous both waysProbabilityRandom Variables and Distribution Functions, Expectation and MomentsHard
- 134Complete convergence implies almost-sure convergence, with no independence neededProbabilityRandom Variables and Distribution Functions, Expectation and MomentsHard
- 135Rates of convergence for sums of iid mean-zero variables — checking each exponent by ChebyshevProbabilityCharacteristic Functions, Probability InequalitiesHard
- 136What convergence in distribution does and does not carry overProbabilityRandom Variables and Distribution Functions, Expectation and MomentsHard
- 137Quadratic-mean convergence — what survives under addition, multiplication and e^i(cdot)ProbabilityCharacteristic Functions, Probability InequalitiesHard
- 138Kolmogorov's 0–1 law — convergence of an independent series is a tail eventProbabilityRandom Variables and Distribution Functions, Expectation and MomentsHard
- 139Building martingales from mean-zero, unit-variance incrementsStochastic ProcessesMartingalesHard
- 140Decomposing a mixed distribution function into its discrete and continuous partsProbabilityRandom Variables and Distribution Functions, Expectation and MomentsHard
- 141Discontinuities of a CDF are always at most countableProbabilityRandom Variables and Distribution Functions, Expectation and MomentsEasy
- 142Jensen's inequality for the (concave) logarithmProbabilityCharacteristic Functions, Probability InequalitiesHard
- 143Lévy's inversion formulaProbabilityCharacteristic Functions, Probability InequalitiesMedium
- 144Bivariate normal conditioning — using the variance bound to rule out two optionsStatistical InferenceMethods of Estimation, Properties of EstimatorsMedium
- 145A loss function engineered to make bar X both Bayes and minimaxStatistical InferenceElementary Bayesian InferenceHard
- 146The Beta family is conjugate for Bernoulli — but the printed option has a typoStatistical InferenceElementary Bayesian InferenceHard
- 147Minimal sufficiency for a symmetric uniform location familyStatistical InferenceMethods of Estimation, Properties of EstimatorsMedium
- 148Unbiasedness for a symmetric uniform location family — checking each candidateStochastic ProcessesMarkov Chains: Classification of States, Stationary DistributionMedium
- 149MLE for the symmetric uniform (-theta,theta) familyStatistical InferenceMethods of Estimation, Properties of EstimatorsEasy
- 150Nested rejection regions across significance levelsStatistical InferenceTests of Hypotheses: MP and UMP Tests, Likelihood Ratio TestsHard
- 151Consistency for a shifted-uniform location parameter — the maximum converges to the wrong pointStatistical InferenceMethods of Estimation, Properties of EstimatorsMedium
- 152Asymptotic normality of bar X/S for exponential data, via the delta methodDistribution TheoryStandard Discrete and Continuous Univariate DistributionsHard
- 153Delta method for four transformations — checking each derivativeDistribution TheorySampling Distributions, Standard Errors, Asymptotic DistributionsHard
- 154Asymptotic relative efficiency of the median versus a scaled mean for Uniform(0,theta)ProbabilityRandom Variables and Distribution Functions, Expectation and MomentsMedium
- 155Asymptotic variance of the empirical distribution function at a pointProbabilityModes of Convergence, Laws of Large Numbers, Central Limit TheoremEasy
- 156Ratio of independent normals is Cauchy — W is not chi-squareStatistical InferenceAnalysis of Discrete Data and Chi-Square Test of Goodness of FitHard
- 157When a linear transform of a spherical normal has uncorrelated componentsMultivariate AnalysisMultivariate Normal Distribution, Wishart Distribution, Quadratic FormsMedium
- 158Summing independent Wishart matrices — degrees of freedom addMultivariate AnalysisMultivariate Normal Distribution, Wishart Distribution, Quadratic FormsMedium
- 159Standardising a Wishart matrix by its scale parameterMultivariate AnalysisMultivariate Normal Distribution, Wishart Distribution, Quadratic FormsEasy
- 160The multivariate normal characteristic functionMultivariate AnalysisMultivariate Normal Distribution, Wishart Distribution, Quadratic FormsEasy
- 161Estimability with four linked expectations — identify the one genuinely non-estimable combinationLinear Models & RegressionGauss-Markov Models, Estimability, Best Linear Unbiased EstimatorsHard
- 162Prediction variance grows with distance from the mean of the regressorsLinear Models & RegressionSimple and Multiple Linear Regression, Logistic RegressionMedium
- 163Regression residuals are correlated and heteroscedastic, againLinear Models & RegressionSimple and Multiple Linear Regression, Logistic RegressionHard
- 164Regression SS and Error SS — uncorrelated by orthogonal projectionsLinear Models & RegressionSimple and Multiple Linear Regression, Logistic RegressionHard
- 165Orthogonal design columns don't automatically make sums of squares uncorrelatedProbabilityRandom Variables and Distribution Functions, Expectation and MomentsHard
- 166Inclusion probability under a two-stage unequal-then-equal designSampling TheorySimple Random, Stratified and Systematic SamplingHard
- 167Regression estimator's efficiency edge is an asymptotic guaranteeSampling TheoryRatio and Regression MethodsMedium
- 168Variance of a sample proportion under SRSWORSampling TheorySimple Random, Stratified and Systematic SamplingEasy
- 169Connectedness pins the C-matrix rank at v-1Design of ExperimentsConnectedness and Orthogonality of Block Designs, BIBDEasy
- 170Fisher's inequality — necessary, not sufficient, for a BIBD to existDesign of ExperimentsConnectedness and Orthogonality of Block Designs, BIBDMedium
- 171Recognising a Youden square from its row patternDesign of ExperimentsCompletely Randomized, Randomized Block and Latin-Square DesignsMedium
- 172Confounding in a replicated 2^3 design — reading the effect off the block layoutDesign of Experiments2^K Factorial Experiments: Confounding and ConstructionHard
- 173Properties of an autocovariance function — spotting the reversed inequalityProbabilityRandom Variables and Distribution Functions, Expectation and MomentsHard
- 174Checking causality and invertibility from the AR and MA characteristic rootsIndex Numbers & Time SeriesTime Series ModelsHard
- 175AR(1) with an intercept — the mean is not simply muProbabilityRandom Variables and Distribution Functions, Expectation and MomentsMedium
- 176Two disjoint closed classes give infinitely many stationary distributionsStochastic ProcessesMarkov Chains: Classification of States, Stationary DistributionHard
- 177Covariance between a Poisson process value and an independent random timeStochastic ProcessesPoisson and Birth-and-Death ProcessesMedium
- 178Extinction probabilities for three branching processesStochastic ProcessesPoisson and Birth-and-Death ProcessesHard
- 179Positive recurrence guarantees a stationary distribution — but not, without irreducibility, a unique oneStochastic ProcessesMarkov Chains: Classification of States, Stationary DistributionHard
- 180The purpose of age-standardisationDemography / Vital StatisticsLife Tables, Mortality RateMedium
- 181Reading the Lx column of a life tableDemography / Vital StatisticsLife Tables, Mortality RateMedium
- 182Average outgoing quality under a two-stage acceptance sampling planStatistical Quality ControlAcceptance SamplingMedium
- 183Matching acceptance-sampling terminology to its definitionStatistical Quality ControlAcceptance SamplingHard
- 184Optimal inventory level for exponential demandOperations ResearchElementary Queuing and Inventory ModelsHard
- 185M/G/1 — waiting time grows with service-time varianceOperations ResearchSteady-State Solutions of Markovian Queuing Models (M/M/1, M/M/C, M/G/1)Easy
- 186A two-node Jackson network — solving the traffic equationsOperations ResearchSteady-State Solutions of Markovian Queuing Models (M/M/1, M/M/C, M/G/1)Hard
- 187Equal allocation minimises a separable sum of squaresOperations ResearchDynamic and Integer ProgrammingMedium
- 188A false claim about dynamic programming's scopeOperations ResearchLinear Programming Problem, Simplex Method, DualityMedium
- 189Constructing the dual of an LPP and checking each constraint's directionOperations ResearchLinear Programming Problem, Simplex Method, DualityMedium
- 190CPM versus PERT — the deterministic/probabilistic distinctionOperations ResearchSequencing and Project SchedulingEasy