December 2020
190 questions, booklet JUN-30220. Booklet labelled June 2020; the sitting was postponed to 27 December 2020.
Section I · General (compulsory) · Q1–10
Practise this section →- 1Which limit-inequality statement survives in full generalityAnalysisSequences and SeriesMedium
- 2Ruling out three properties of a strictly increasing function via its derivativeAnalysisContinuity, Uniform Continuity, Differentiability, Mean Value TheoremEasy
- 3The two-branch characterization of zw, z+winRComplex AnalysisAlgebra of Complex Numbers, Power Series, Transcendental FunctionsEasy
- 4Re-centering the geometric series for 1/z at z=1Complex AnalysisAlgebra of Complex Numbers, Power Series, Transcendental FunctionsEasy
- 5The matrix size of a linear map is codomain-by-domainLinear AlgebraAlgebra of Linear TransformationsEasy
- 6The one false statement about an eigenvalue's defining kernelLinear AlgebraVector Spaces, Subspaces, Linear Dependence, Basis, DimensionMedium
- 7Normalizing a quadratic density and integrating past x=1ProbabilityRandom Variables and Distribution Functions, Expectation and MomentsEasy
- 8Recognising the MGF of a Poisson distributionDistribution TheoryStandard Discrete and Continuous Univariate DistributionsEasy
- 9Spotting alternate optima when the objective is parallel to a binding constraintOperations ResearchLinear Programming Problem, Simplex Method, DualityMedium
- 10Existence of a basic feasible solution is guaranteed; uniqueness of representation is notOperations ResearchLinear Programming Problem, Simplex Method, DualityMedium
Section II · Mathematics · Q11–100
Practise this section →- 11The min-of-maxes formula for the median of three numbersOperations ResearchLinear Programming Problem, Simplex Method, DualityEasy
- 12Every real sequence has a monotone subsequenceAnalysisSequences and SeriesMedium
- 13An open interval can be mapped continuously onto a closed one, never the reverseTopologyBasis, Dense Sets, Subspace and Product TopologyMedium
- 14xsin(1/x) is continuous but nowhere of bounded variation near 0, and not differentiable thereAnalysisSequences and SeriesMedium
- 15Every set is clopen and bounded in the discrete metric — only infinite sets fail to be compactTopologySeparation Axioms, Connectedness and CompactnessMedium
- 16Closer to i than to -i means lying in the upper half planeComplex AnalysisAlgebra of Complex Numbers, Power Series, Transcendental FunctionsEasy
- 17The strip 0<Re(z)lepi/2 maps onto the right half plane under sin zComplex AnalysisAlgebra of Complex Numbers, Power Series, Transcendental FunctionsMedium
- 18A Liouville-type theorem for non-negative harmonic functions, and logz's missing conjugateComplex AnalysisAnalytic Functions, Cauchy-Riemann EquationsMedium
- 19Extracting a residue from the Laurent series of cos(1/z)Complex AnalysisTaylor Series, Laurent Series, Calculus of ResiduesEasy
- 20A third-order pole hiding behind an apparent double zeroComplex AnalysisTaylor Series, Laurent Series, Calculus of ResiduesEasy
- 21Homomorphic images of cyclic groups stay cyclic — but the converse failsAlgebraGroups, Subgroups, Sylow Theorems, Cayley's TheoremEasy
- 22X^2-n is irreducible for squarefree n1, giving a field — hence, in particular, a domainAlgebraRings, Ideals, UFD, PID, Euclidean DomainEasy
- 23Cayley's theoremAlgebraGroups, Subgroups, Sylow Theorems, Cayley's TheoremEasy
- 24The order of a permutation is the lcm of its disjoint cycle lengthsAlgebraPermutations, Combinations, Pigeon-Hole Principle, Inclusion-ExclusionMedium
- 25Recognising when a two-generator ideal collapses to a principal oneAlgebraRings, Ideals, UFD, PID, Euclidean DomainMedium
- 26Spotting linearity: no constant shifts, no products of coordinatesLinear AlgebraAlgebra of Linear TransformationsMedium
- 27A minimal polynomial that splits over R guarantees triangulability, not diagonalizabilityLinear AlgebraEigenvalues and Eigenvectors, Cayley-Hamilton TheoremMedium
- 28The rank–nullity theorem, restated for the quotient spaceLinear AlgebraVector Spaces, Subspaces, Linear Dependence, Basis, DimensionEasy
- 29Reading a linear operator's matrix off its coordinate formulasLinear AlgebraVector Spaces, Subspaces, Linear Dependence, Basis, DimensionMedium
- 30Same rank alone does not force congruence — Sylvester's law of inertia also needs the signatureLinear AlgebraAlgebra of Matrices, Rank, Determinant, Linear EquationsMedium
- 31Recognising a semilinear PDE: linear principal part, nonlinear lower-order termAnalysisContinuity, Uniform Continuity, Differentiability, Mean Value TheoremEasy
- 32The complete integral of f(p,q)=0Partial Differential EquationsSeparation of Variables (Laplace, Heat and Wave Equations)Easy
- 33A vanishing Wronskian that does not signal linear dependenceLinear AlgebraVector Spaces, Subspaces, Linear Dependence, Basis, DimensionMedium
- 34Solving x,dy-y,dx=0: the family of lines through the originOrdinary Differential EquationsGeneral Theory of Linear ODEs, Variation of Parameters, Sturm-Liouville, Green's FunctionEasy
- 35Clearing a fractional power before reading off the degreeOrdinary Differential EquationsGeneral Theory of Linear ODEs, Variation of Parameters, Sturm-Liouville, Green's FunctionEasy
- 36A degenerate North-West corner allocation, verified by direct cost computationOperations ResearchTransportation and Assignment ProblemsHard
- 37Feasible yet unbounded: a genuinely combined-letter key answerOperations ResearchLinear Programming Problem, Simplex Method, DualityMedium
- 38The dual of the dual is the primal; not every feasible point is basicOperations ResearchLinear Programming Problem, Simplex Method, DualityEasy
- 39Constructing the dual and matching coefficients term by termOperations ResearchLinear Programming Problem, Simplex Method, DualityMedium
- 40An optimum can exist on an unbounded feasible regionOperations ResearchLinear Programming Problem, Simplex Method, DualityHard
- 41An odd integrand over a centrally symmetric slice vanishesAnalysisMonotonic Functions, Bounded Variation, Lebesgue Measure & IntegralEasy
- 42A rank-deficient linear map cannot be an open map into the full codomainAnalysisContinuity, Uniform Continuity, Differentiability, Mean Value TheoremHard
- 43A vector orthogonal to the gradientLinear AlgebraInner Product Spaces, Orthonormal BasisEasy
- 44The open mapping theorem rules out an image that is merely a circleAnalysisMetric Spaces, Compactness, Connectedness, Normed Linear SpacesMedium
- 45Two exponentials, linearly independent, forced to vanish identicallyLinear AlgebraVector Spaces, Subspaces, Linear Dependence, Basis, DimensionEasy
- 46A strict derivative bound forces exponential form, and hence f"=cf'AnalysisContinuity, Uniform Continuity, Differentiability, Mean Value TheoremMedium
- 47Counting maximal ideals via the irreducible factorization of X^8+1 over RComplex AnalysisAlgebra of Complex Numbers, Power Series, Transcendental FunctionsEasy
- 48Z[sqrt-p] fails unique factorization for every odd prime pAlgebraRings, Ideals, UFD, PID, Euclidean DomainMedium
- 49The ring of analytic functions is a non-Noetherian integral domainAnalysisSequences and SeriesHard
- 50(-1,1) is completely metrizable even though its usual metric is not completeAnalysisMetric Spaces, Compactness, Connectedness, Normed Linear SpacesHard
- 51An unbounded hyperbola pair versus the compact ell^1-unit circleAnalysisMetric Spaces, Compactness, Connectedness, Normed Linear SpacesEasy
- 52Inner regularity forces measure zero, which gives outer-regular open covers of arbitrarily small measureAnalysisMonotonic Functions, Bounded Variation, Lebesgue Measure & IntegralMedium
- 53Every prime-power-order group is solvable, even though it need not be abelian or simpleAlgebraGroups, Subgroups, Sylow Theorems, Cayley's TheoremMedium
- 54The classic non-Noetherian ring: polynomials in infinitely many variablesLinear AlgebraVector Spaces, Subspaces, Linear Dependence, Basis, DimensionMedium
- 55Counting intermediate fields via the subgroup lattice of S3AlgebraGroups, Subgroups, Sylow Theorems, Cayley's TheoremEasy
- 56The Open Mapping Theorem forces both pullback norms to be equivalent to the originalsAnalysisMetric Spaces, Compactness, Connectedness, Normed Linear SpacesHard
- 57A co-isometry that fails every one of normal, self-adjoint, unitary, and T^T=IAnalysisMetric Spaces, Compactness, Connectedness, Normed Linear SpacesHard
- 58Integration is a bounded operator on C[0,1]; differentiation, under the sup norm, is notAnalysisContinuity, Uniform Continuity, Differentiability, Mean Value TheoremHard
- 59Only the closure containment survives for a general continuous map; interior and boundary containments can failTopologyBasis, Dense Sets, Subspace and Product TopologyHard
- 60Two components of an open disc each, regardless of gap size or how the split arises, are all homeomorphicTopologySeparation Axioms, Connectedness and CompactnessHard
- 61Every subfield of R contains Q, hence is dense — so continuity forces the zero functionAnalysisContinuity, Uniform Continuity, Differentiability, Mean Value TheoremMedium
- 62A lattice's ideals are all prime exactly when the lattice is totally orderedAlgebraRings, Ideals, UFD, PID, Euclidean DomainEasy
- 63A tree with ge3 vertices is its own unique spanning treeGraph TheoryGraphs, Trees, ConnectivityHard
- 64Pigeonhole: one more than the number of couples forces a full pairAlgebraPermutations, Combinations, Pigeon-Hole Principle, Inclusion-ExclusionEasy
- 65Uniqueness for the Dirichlet problem via the maximum principlePartial Differential EquationsSeparation of Variables (Laplace, Heat and Wave Equations)Medium
- 66The cosine Fourier series for the even function xIntegral TransformsFourier TransformHard
- 67PDE type is a coordinate-independent invariantPartial Differential EquationsClassification of Second Order PDEsMedium
- 68Infinitely many integer solutions of 7x-8y=5, spaced 8 apart in xAlgebraFundamental Theorem of Arithmetic, Congruences, Chinese Remainder TheoremMedium
- 69Spotting 13^2equiv40pmod43 among the given optionsComplex AnalysisTaylor Series, Laurent Series, Calculus of ResiduesEasy
- 70One of any four consecutive integers is divisible by 4, killing the Mobius productComplex AnalysisConformal Mappings, Mobius TransformationsEasy
- 71Three rigidly-linked particles in space have the 6 degrees of freedom of a rigid bodyClassical MechanicsMotion of Rigid Bodies, Euler's Dynamical Equations, Small OscillationsMedium
- 72Kinetic energy in plane polar coordinatesClassical MechanicsMotion of Rigid Bodies, Euler's Dynamical Equations, Small OscillationsEasy
- 73The harmonic-oscillator Hamiltonian gives angular frequency k, not k^2Classical MechanicsHamilton's Canonical Equations, Hamilton's PrincipleEasy
- 74The inertia tensor connects two genuinely distinct physical vectorsLinear AlgebraAlgebra of Linear TransformationsHard
- 75Steady flow makes all three particle-tracking curves coincideFluid DynamicsPotential Flow, Streamlines, DoubletsMedium
- 76The standard sufficient conditions for a velocity potential to existFluid DynamicsPotential Flow, Streamlines, DoubletsEasy
- 77Streamlines from a velocity potential via dx/u=dy/vFluid DynamicsPotential Flow, Streamlines, DoubletsEasy
- 78The classical Rankine half-body: a stagnation point genuinely upstream of the sourceFluid DynamicsPotential Flow, Streamlines, DoubletsHard
- 79The first fundamental form is always non-singular; the second need not beLinear AlgebraAlgebra of Matrices, Rank, Determinant, Linear EquationsMedium
- 80A unit first-fundamental-form determinant means area is preserved, not necessarily anglesDifferential GeometryCurves and Surfaces, CurvatureMedium
- 81A local max and a local min of a graph function both give non-negative Gaussian curvatureLinear AlgebraAlgebra of Matrices, Rank, Determinant, Linear EquationsMedium
- 82The extremal family for int(y^2-y'^2)dx, satisfying one endpoint conditionCalculus of VariationsVariation of a Functional, Euler-Lagrange Equation, Conditions for ExtremaEasy
- 83Computing the first variation term by termCalculus of VariationsVariation of a Functional, Euler-Lagrange Equation, Conditions for ExtremaMedium
- 84Beltrami's identity for an x-free integrand gives a hyperbolic-sine familyCalculus of VariationsVariation of a Functional, Euler-Lagrange Equation, Conditions for ExtremaHard
- 85Minimizing the distance from a point on y=x^2 to the line x-y=5AnalysisContinuity, Uniform Continuity, Differentiability, Mean Value TheoremEasy
- 86Non-vanishing Fredholm determinant forces the trivial solution aloneLinear Integral EquationsFredholm and Volterra Type, Separable Kernels, Resolvent KernelEasy
- 87Solving a separable-kernel eigenvalue problem for k(t,s)=e^t+sLinear Integral EquationsFredholm and Volterra Type, Separable Kernels, Resolvent KernelEasy
- 88Solving a convolution-type Volterra equation of the first kind via Laplace transformsLinear Integral EquationsFredholm and Volterra Type, Separable Kernels, Resolvent KernelEasy
- 89Testing the standard finite-difference operator identitiesNumerical AnalysisFinite Differences, Lagrange/Hermite/Spline InterpolationMedium
- 90Recovering a missing table value by forcing the fourth difference to vanishNumerical AnalysisFinite Differences, Lagrange/Hermite/Spline InterpolationEasy
- 91The standard Trapezoidal ruleNumerical AnalysisNumerical Differentiation and IntegrationMedium
- 92The Laplace transform of the zero-order Bessel functionIntegral TransformsLaplace TransformEasy
- 93The Fourier cosine transform of a Gaussian is another GaussianIntegral TransformsFourier TransformEasy
- 94The convolution theorem for Laplace transformsIntegral TransformsLaplace TransformEasy
- 95Building the Lagrangian with squared slack variables for inequality constraintsOperations ResearchLinear Programming Problem, Simplex Method, DualityHard
- 96Inclusion–exclusion for integers avoiding three small prime divisorsAlgebraPermutations, Combinations, Pigeon-Hole Principle, Inclusion-ExclusionEasy
- 97Chartrand's theorem: connectivity, edge connectivity and minimum degree coincide for cubic graphsGraph TheoryGraphs, Trees, ConnectivityEasy
- 98An integral against a sign-changing density is a signed measure, not a measureAnalysisMonotonic Functions, Bounded Variation, Lebesgue Measure & IntegralMedium
- 99The intersection of a positive and a negative set is always nu-nullAnalysisMonotonic Functions, Bounded Variation, Lebesgue Measure & IntegralEasy
- 100The limit points of a dense set fill up the whole lineTopologyBasis, Dense Sets, Subspace and Product TopologyMedium
Section III · Statistics · Q101–190
Practise this section →- 101Mean median mode signals right skewDescriptive StatisticsDescriptive Statistics, Exploratory Data AnalysisEasy
- 102Correlation as the geometric mean of the two regression slopes, sign includedLinear Models & RegressionSimple and Multiple Linear Regression, Logistic RegressionEasy
- 103Symmetric tail probabilities around the mean need no explicit sigmaDistribution TheoryStandard Discrete and Continuous Univariate DistributionsEasy
- 104E[X^2]<(E[X])^2 would require negative varianceProbabilityCharacteristic Functions, Probability InequalitiesEasy
- 105Disjoint events are never (nondegenerately) independentProbabilityRandom Variables and Distribution Functions, Expectation and MomentsMedium
- 106E[XYmid X]=X, not the constant 1, for independent unit-mean X,YProbabilityCharacteristic Functions, Probability InequalitiesMedium
- 107Reading a joint pmf table directly to find P[X=2Y]ProbabilityRandom Variables and Distribution Functions, Expectation and MomentsHard
- 108A covariance matrix must be symmetric and positive semi-definiteMultivariate AnalysisMultivariate Normal Distribution, Wishart Distribution, Quadratic FormsMedium
- 109The reciprocal of a Cauchy is Cauchy, with a location/scale inversion formulaProbabilityRandom Variables and Distribution Functions, Expectation and MomentsMedium
- 110Poisson probability of at least one error in three times the base intervalProbabilityRandom Variables and Distribution Functions, Expectation and MomentsEasy
- 111Constant hazard rate characterizes the exponential distributionDistribution TheoryStandard Discrete and Continuous Univariate DistributionsMedium
- 112A linear combination of iid normal vectors by row-vector weights is scalar (univariate) normalDistribution TheoryStandard Discrete and Continuous Univariate DistributionsEasy
- 113The extreme order statistics are jointly sufficient for a fixed-width uniform location parameterDistribution TheoryDistribution of Order Statistics and RangeMedium
- 114Coverage probability of the sample range as a confidence interval, and a printed-interval flagStatistical InferenceConfidence IntervalsHard
- 115Matching the chi^2 degrees of freedom to the estimated meanStatistical InferenceTests of Hypotheses: MP and UMP Tests, Likelihood Ratio TestsHard
- 116The Cramer–Rao bound is a lower bound restricted to unbiased estimatorsStatistical InferenceMethods of Estimation, Properties of EstimatorsMedium
- 117The pooled sums and combined sum of squares form a complete sufficient statistic for two normal samples sharing sigma^2Statistical InferenceMethods of Estimation, Properties of EstimatorsHard
- 118The F-test for equality of variances needs both independence and normalityStatistical InferenceTests of Hypotheses: MP and UMP Tests, Likelihood Ratio TestsHard
- 119A pivotal quantity's distribution is fully known and parameter-freeStatistical InferenceConfidence IntervalsMedium
- 120The standard one-sample proportion test statisticStatistical InferenceTests of Hypotheses: MP and UMP Tests, Likelihood Ratio TestsMedium
- 121Degrees of freedom for testing homogeneity of r multinomial populationsStatistical InferenceTests of Hypotheses: MP and UMP Tests, Likelihood Ratio TestsEasy
- 122Reading the Mann–Whitney U statistic off a fully specified combined rankingStatistical InferenceNonparametric Tests, Rank Correlation, Test for IndependenceMedium
- 123Sequencing a two-machine flow shop by Johnson's ruleOperations ResearchSequencing and Project SchedulingHard
- 124Re-order level sits above minimum (safety) stock by constructionOperations ResearchElementary Queuing and Inventory ModelsEasy
- 125Doubling both arrival and service rates in M/M/1 halves the average waitOperations ResearchSteady-State Solutions of Markovian Queuing Models (M/M/1, M/M/C, M/G/1)Medium
- 126A mixed selection mechanism gives multiples of k two routes into the sample, breaking equal inclusion probabilitySampling TheorySimple Random, Stratified and Systematic SamplingHard
- 127Systematic sampling is efficient precisely when the population carries a linear trendSampling TheorySimple Random, Stratified and Systematic SamplingHard
- 128Estimability in one-way ANOVA: the coefficient of mu must equal the sum of the alpha coefficientsLinear Models & RegressionGauss-Markov Models, Estimability, Best Linear Unbiased EstimatorsMedium
- 129Common-variance contrasts hold generally, but not every pairwise BLUE pair is uncorrelatedDesign of ExperimentsCompletely Randomized, Randomized Block and Latin-Square DesignsHard
- 130Total degrees of freedom in a two-replicate 2^3 design are not simply twice the treatment degrees of freedomDesign of ExperimentsCompletely Randomized, Randomized Block and Latin-Square DesignsHard
- 131Most uncountable subsets of R fail to be BorelAnalysisMonotonic Functions, Bounded Variation, Lebesgue Measure & IntegralMedium
- 132A crude finite/infinite indicator set function fails finite additivityProbabilityRandom Variables and Distribution Functions, Expectation and MomentsHard
- 133The exponential measure is absolutely continuous w.r.t. Lebesgue measure, never the reverseProbabilityRandom Variables and Distribution Functions, Expectation and MomentsHard
- 134Products and exponentials of an a.s.-vanishing sequence stay degenerate at their limitsProbabilityModes of Convergence, Laws of Large Numbers, Central Limit TheoremHard
- 135Two limits that quote the wrong constant: E[X^2] and Var(X), not 2 and 0ProbabilityRandom Variables and Distribution Functions, Expectation and MomentsHard
- 136Almost-sure convergence transfers through bounded continuous maps, but not through the unbounded exponentialProbabilityRandom Variables and Distribution Functions, Expectation and MomentsHard
- 137Uniform boundedness with diverging total variance forces the CLT via Lyapunov's conditionProbabilityModes of Convergence, Laws of Large Numbers, Central Limit TheoremHard
- 138Kolmogorov's convergence theorem: summable variances force a.s. convergenceProbabilityRandom Variables and Distribution Functions, Expectation and MomentsEasy
- 139A recursively-defined process can also be a martingale, but adding a deterministic drift breaks itStochastic ProcessesMartingalesHard
- 140Computing E(X) for a mixed distribution: two atoms plus a continuous pieceProbabilityRandom Variables and Distribution Functions, Expectation and MomentsMedium
- 141A monotone function has at most countably many jumps, leaving uncountably many continuity pointsProbabilityRandom Variables and Distribution Functions, Expectation and MomentsEasy
- 142The Minkowski inequality for the L^3 normProbabilityRandom Variables and Distribution Functions, Expectation and MomentsEasy
- 143Fourier inversion recovers the density from an integrable characteristic functionProbabilityCharacteristic Functions, Probability InequalitiesEasy
- 144The law of total covariance shows X and Y share only a fraction of their variability — never perfect correlationProbabilityRandom Variables and Distribution Functions, Expectation and MomentsMedium
- 145Mixed moments of a multivariate normal from derivatives of its characteristic functionMultivariate AnalysisMultivariate Normal Distribution, Wishart Distribution, Quadratic FormsHard
- 146The likelihood's truncation collapses the posterior to a bounded uniformStatistical InferenceElementary Bayesian InferenceMedium
- 147Minimal sufficiency for a family of uniforms with index-scaled supportStatistical InferenceMethods of Estimation, Properties of EstimatorsMedium
- 148The MLE beats the UMVUE in MSE for normal variance estimationStatistical InferenceMethods of Estimation, Properties of EstimatorsHard
- 149The score function has mean zero under standard regularityProbabilityRandom Variables and Distribution Functions, Expectation and MomentsEasy
- 150Monotone likelihood ratio in bar X gives the UMP upper-tail Poisson testStatistical InferenceTests of Hypotheses: MP and UMP Tests, Likelihood Ratio TestsMedium
- 1512bar Xn, not bar Xn itself, is consistent for the uniform upper endpointStatistical InferenceMethods of Estimation, Properties of EstimatorsMedium
- 152Asymptotic normality of an order statistic near the first quartile of a Pareto-type distributionDistribution TheorySampling Distributions, Standard Errors, Asymptotic DistributionsHard
- 153Delta method applied to logbar X for a Gamma-distributed sampleDistribution TheorySampling Distributions, Standard Errors, Asymptotic DistributionsMedium
- 154At n=2 the sample mean and the midrange are literally the same statisticDistribution TheoryDistribution of Order Statistics and RangeEasy
- 155Deriving the Fisher z-transformation from the delta methodDistribution TheorySampling Distributions, Standard Errors, Asymptotic DistributionsMedium
- 156Principal components share independence but not a common varianceMultivariate AnalysisMultivariate Normal Distribution, Wishart Distribution, Quadratic FormsHard
- 157Eigenvector rows diagonalize Sigma, guaranteeing independent marginalsProbabilityRandom Variables and Distribution Functions, Expectation and MomentsMedium
- 158An idempotent-matrix quadratic form in Mahalanobis coordinates gives a chi-square with rank-many degrees of freedomMultivariate AnalysisMultivariate Normal Distribution, Wishart Distribution, Quadratic FormsEasy
- 159The standard multivariate-normal conditional mean and covariance formulasProbabilityRandom Variables and Distribution Functions, Expectation and MomentsHard
- 160Cochran's theorem: rank-sum equal to n plus positive semi-definiteness gives independent chi-squaresStatistical InferenceAnalysis of Discrete Data and Chi-Square Test of Goodness of FitHard
- 161Solving the normal equations for an exactly-identified three-parameter, three-observation modelProbabilityRandom Variables and Distribution Functions, Expectation and MomentsHard
- 162The least-squares line passes through (bar x,bar y) exactly when an intercept is fittedLinear Models & RegressionSimple and Multiple Linear Regression, Logistic RegressionMedium
- 163Uncorrelated errors give non-identical, uncorrelated responsesLinear Models & RegressionSimple and Multiple Linear Regression, Logistic RegressionMedium
- 164Two standard OLS orthogonality identities: residuals sum to zero and are orthogonal to fitted valuesDesign of ExperimentsConnectedness and Orthogonality of Block Designs, BIBDMedium
- 165Testing a variance component's insignificance, not a mean-equality hypothesis, for random effectsLinear Models & RegressionAnalysis of Variance and Covariance, Fixed/Random/Mixed EffectsMedium
- 166Only one of the two possible systematic samples contains both unitsSampling TheorySimple Random, Stratified and Systematic SamplingEasy
- 167Neyman allocation minimizes sample size for a fixed target varianceOperations ResearchLinear Programming Problem, Simplex Method, DualityHard
- 168Post-stratification approaches full stratification's efficiency when strata are well-filledSampling TheorySimple Random, Stratified and Systematic SamplingHard
- 169Counting estimable orthogonal contrasts via the rank of the block design's information matrixDesign of ExperimentsConnectedness and Orthogonality of Block Designs, BIBDHard
- 170Deriving k,lambda for a symmetric BIBD from the two standard identitiesDesign of ExperimentsConnectedness and Orthogonality of Block Designs, BIBDEasy
- 171No listed option correctly caps the M/M/K departure rate at kmuOperations ResearchSteady-State Solutions of Markovian Queuing Models (M/M/1, M/M/C, M/G/1)Hard
- 172Identifying the confounded effect from a 2^4 block split via sign parityDesign of Experiments2^K Factorial Experiments: Confounding and ConstructionMedium
- 173The one-step-ahead AR(1) forecast-error ratio is always strictly positiveIndex Numbers & Time SeriesTime Series ModelsHard
- 174Variance of the mean of ten equicorrelated unit-variance variablesProbabilityRandom Variables and Distribution Functions, Expectation and MomentsHard
- 175Only one candidate MSE formula wrongly tacks on an extra mu^2 termProbabilityRandom Variables and Distribution Functions, Expectation and MomentsHard
- 176Marginalizing a mixed initial distribution through one step of the transition matrixStochastic ProcessesMarkov Chains: Classification of States, Stationary DistributionEasy
- 177Conditional on the Poisson count, event times are order statistics of a uniform sampleStochastic ProcessesPoisson and Birth-and-Death ProcessesEasy
- 178Summing a geometric series of branching-process generation meansStochastic ProcessesPoisson and Birth-and-Death ProcessesEasy
- 179A guaranteed but finite-mean return to 0 makes every state positive recurrentStochastic ProcessesMarkov Chains: Classification of States, Stationary DistributionHard
- 180The denominator of the infant mortality rateDemography / Vital StatisticsLife Tables, Mortality RateEasy
- 181Each parent's own survival probability is conditioned on their own starting ageDemography / Vital StatisticsLife Tables, Mortality RateHard
- 182Estimating process standard deviation from the average rangeStatistical Quality ControlControl ChartsEasy
- 183Adding failure rates for a series system gives a bathtub-shaped, non-monotone hazardSurvival AnalysisHazard Function and Failure Rates, Censoring and Life TestingMedium
- 184Direct substitution into the inventory-position formulaOperations ResearchElementary Queuing and Inventory ModelsEasy
- 185Reserving servers for a customer category is control of the queue disciplineOperations ResearchElementary Queuing and Inventory ModelsMedium
- 186Both standard interpretations of the Lindley recursion's variables are validOperations ResearchElementary Queuing and Inventory ModelsMedium
- 187Bellman's principle guarantees optimal subpaths, but not their uniquenessOperations ResearchDynamic and Integer ProgrammingMedium
- 188AM–GM gives the minimum of y1^2+y2^2 under a fixed productOperations ResearchDynamic and Integer ProgrammingEasy
- 189Karmarkar's canonical form requires homogeneous, not non-homogeneous, constraintsOperations ResearchLinear Programming Problem, Simplex Method, DualityMedium
- 190Series-system failure rates add for independent componentsSurvival AnalysisHazard Function and Failure Rates, Censoring and Life TestingEasy