June 2019
190 questions, booklet JUN-30219.
Section I · General (compulsory) · Q1–10
Practise this section →- 1Only the closed shrinking intervals share a common pointAnalysisElementary Set Theory, Real Number System, Supremum, InfimumEasy
- 2sin x-sin ylex-y via the mean value theoremAnalysisContinuity, Uniform Continuity, Differentiability, Mean Value TheoremEasy
- 3The positive subsequence (n+1)/n decreases to 1, giving the limsupAnalysisSequences and SeriesEasy
- 4Evaluating i^i via the principal logarithmComplex AnalysisAlgebra of Complex Numbers, Power Series, Transcendental FunctionsEasy
- 5Doubling real dimension when a complex space is viewed over RLinear AlgebraVector Spaces, Subspaces, Linear Dependence, Basis, DimensionEasy
- 6A basis-shift-down operator is nilpotent of index nLinear AlgebraVector Spaces, Subspaces, Linear Dependence, Basis, DimensionMedium
- 7A binomial probability with a 2:1 biased coinDistribution TheoryStandard Discrete and Continuous Univariate DistributionsEasy
- 8Mutual exclusivity makes "A but not B" just P(A)ProbabilitySample Space, Discrete Probability, Independent Events, Bayes TheoremEasy
- 9Locating the optimum along a shared-value edge of the feasible polygonOperations ResearchLinear Programming Problem, Simplex Method, DualityMedium
- 10The fundamental theorem of linear programmingOperations ResearchLinear Programming Problem, Simplex Method, DualityMedium
Section II · Mathematics · Q11–100
Practise this section →- 11The supremum of a union is the max of the two supremaAnalysisElementary Set Theory, Real Number System, Supremum, InfimumEasy
- 12Differentiating a self-mirrored integral identity forces fequiv0AnalysisRiemann Sums and Riemann Integral, Improper IntegralsEasy
- 13Thomae's function: Riemann integrable, continuous at irrationals, discontinuous at rationalsAnalysisMonotonic Functions, Bounded Variation, Lebesgue Measure & IntegralEasy
- 14A differentiable function with bounded, oscillating derivative is still of bounded variationAnalysisSequences and SeriesMedium
- 15Compactness, not mere closedness, guarantees positive distance to a disjoint closed setAnalysisMetric Spaces, Compactness, Connectedness, Normed Linear SpacesMedium
- 16Three points determine a Mobius map, and there are infinitely many valid choicesComplex AnalysisConformal Mappings, Mobius TransformationsEasy
- 17e^z= const describes a vertical lineAnalysisElementary Set Theory, Real Number System, Supremum, InfimumEasy
- 18Cauchy–Riemann equations hold only on the real axis, so f is analytic nowhereAnalysisContinuity, Uniform Continuity, Differentiability, Mean Value TheoremEasy
- 19A doubly-wound contour picks up twice the residueComplex AnalysisTaylor Series, Laurent Series, Calculus of ResiduesEasy
- 20Poles of csc(1/(z-1)) accumulate at z=1 itselfComplex AnalysisTaylor Series, Laurent Series, Calculus of ResiduesEasy
- 21Pairing each element with its inverse forces an odd count of involutionsAlgebraGroups, Subgroups, Sylow Theorems, Cayley's TheoremEasy
- 22Every proper-subgroup quotient of (Q,+) is infiniteAlgebraGroups, Subgroups, Sylow Theorems, Cayley's TheoremEasy
- 23A prime-order group is cyclic with only the trivial subgroupsAlgebraGroups, Subgroups, Sylow Theorems, Cayley's TheoremEasy
- 24Counting GL(2,mathbf Z2) directlyAlgebraGroups, Subgroups, Sylow Theorems, Cayley's TheoremEasy
- 25A one-variable polynomial ring over a field is always a PIDAlgebraRings, Ideals, UFD, PID, Euclidean DomainEasy
- 26Detecting a hidden linear dependence among the spanning vectorsLinear AlgebraVector Spaces, Subspaces, Linear Dependence, Basis, DimensionEasy
- 27Only the standard Hermitian form is positive definite on mathbf C^2Linear AlgebraInner Product Spaces, Orthonormal BasisEasy
- 28Reading a linear map's matrix off its coordinate formulasLinear AlgebraAlgebra of Linear TransformationsEasy
- 29Nonzero eigenvalues (real or complex) already force invertibilityLinear AlgebraAlgebra of Matrices, Rank, Determinant, Linear EquationsEasy
- 30The adjoint's matrix is the conjugate transposeLinear AlgebraVector Spaces, Subspaces, Linear Dependence, Basis, DimensionMedium
- 31The defining PDE for axial symmetry about the z-axisPartial Differential EquationsLagrange and Charpit Methods, Cauchy Problem for First Order PDEsMedium
- 32Factoring the wave operator partialx^2-partialy^2Partial Differential EquationsSeparation of Variables (Laplace, Heat and Wave Equations)Easy
- 33The solution space of an nth order linear ODE is exactly n-dimensionalLinear AlgebraVector Spaces, Subspaces, Linear Dependence, Basis, DimensionEasy
- 34Solving a Bernoulli equation via the substitution v=1/yOrdinary Differential EquationsExistence & Uniqueness, Singular Solutions, Systems of First Order ODEsEasy
- 35Bounded partial derivatives give Lipschitz; a fractional power in y alone does notAnalysisContinuity, Uniform Continuity, Differentiability, Mean Value TheoremMedium
- 36Comparing the four proposed assignments directly by total costOperations ResearchTransportation and Assignment ProblemsEasy
- 37An equality constraint pins x1, forcing two remaining constraints into direct contradictionOperations ResearchLinear Programming Problem, Simplex Method, DualityEasy
- 38Both halves of the standard LP duality correspondence holdOperations ResearchLinear Programming Problem, Simplex Method, DualityMedium
- 39Constructing the dual and matching every coefficientOperations ResearchLinear Programming Problem, Simplex Method, DualityMedium
- 40Weak duality: an unbounded dual forces primal infeasibilityOperations ResearchLinear Programming Problem, Simplex Method, DualityEasy
- 41Compactness of [a,b] forces its continuous image to be boundedAnalysisContinuity, Uniform Continuity, Differentiability, Mean Value TheoremEasy
- 42Squaring a complex number is injective away from antipodal pairsComplex AnalysisConformal Mappings, Mobius TransformationsMedium
- 43Composing two Jacobians via the chain ruleLinear AlgebraAlgebra of Matrices, Rank, Determinant, Linear EquationsMedium
- 44Only linear polynomials give bijective entire functionsComplex AnalysisLiouville's Theorem, Maximum Modulus Principle, Open Mapping TheoremEasy
- 45The open mapping theorem rules out a closed-disc imageAnalysisMetric Spaces, Compactness, Connectedness, Normed Linear SpacesEasy
- 46Choosing g=-f+c makes bar f+bar g collapse to a constantComplex AnalysisAlgebra of Complex Numbers, Power Series, Transcendental FunctionsHard
- 47The unit group of mathbf Fp is simple exactly when p-1 is primeAlgebraFields, Finite Fields, Field Extensions, Galois TheoryMedium
- 48The classic irreducible-but-not-prime element on the coordinate ring of the circleAlgebraRings, Ideals, UFD, PID, Euclidean DomainEasy
- 49The trace form for C/R reduces to 2Re(z^2)Linear AlgebraQuadratic Forms, Reduction and ClassificationMedium
- 50Completeness makes R second category; countability makes Q first categoryAnalysisMetric Spaces, Compactness, Connectedness, Normed Linear SpacesEasy
- 51Countable sets never move Lebesgue measureAnalysisMonotonic Functions, Bounded Variation, Lebesgue Measure & IntegralMedium
- 52The complement of a nowhere dense set is always denseAnalysisMetric Spaces, Compactness, Connectedness, Normed Linear SpacesMedium
- 53Order 12 is the smallest of the four options admitting non-abelian examplesAlgebraGroups, Subgroups, Sylow Theorems, Cayley's TheoremEasy
- 54Z72congZ9timesZ8 by the Chinese Remainder TheoremAlgebraFundamental Theorem of Arithmetic, Congruences, Chinese Remainder TheoremEasy
- 55Finite field extensions are always cyclic Galois, via the FrobeniusAlgebraFields, Finite Fields, Field Extensions, Galois TheoryMedium
- 56Stirling's growth versus a root that tames a logarithmAnalysisSequences and SeriesEasy
- 57Uncountably many pm1 sign patterns give unitary diagonal operatorsAnalysisSequences and SeriesMedium
- 58Orthogonal halves of C[0,1] that cannot rebuild a function non-zero at 1/2AnalysisMetric Spaces, Compactness, Connectedness, Normed Linear SpacesHard
- 59Closure in the upper-ray topology: everything not strictly aboveTopologyBasis, Dense Sets, Subspace and Product TopologyEasy
- 60Two simple closed curves are homeomorphic to each other, but not to a branched "X"TopologySeparation Axioms, Connectedness and CompactnessMedium
- 61Continuous open surjections push forward a countable baseAnalysisElementary Set Theory, Real Number System, Supremum, InfimumEasy
- 62Completeness gives boundedness, and Boolean lattices are always distributiveAlgebraLattice Theory (Distributive/Modular/Boolean Lattices)Medium
- 63Every-even-degree graphs decompose into edge-disjoint cyclesGraph TheoryGraphs, Trees, ConnectivityEasy
- 64Counting alternating arrangements out of all permutationsAlgebraPermutations, Combinations, Pigeon-Hole Principle, Inclusion-ExclusionEasy
- 65A mean-value computation whose printed option is ambiguously typesetAnalysisContinuity, Uniform Continuity, Differentiability, Mean Value TheoremHard
- 66A negative discriminant from the Laplacian's principal part gives elliptic typePartial Differential EquationsClassification of Second Order PDEsEasy
- 67Uniqueness for the wave equation's Cauchy problemOrdinary Differential EquationsExistence & Uniqueness, Singular Solutions, Systems of First Order ODEsEasy
- 68Squares mod 3 are exactly 0,1AlgebraFundamental Theorem of Arithmetic, Congruences, Chinese Remainder TheoremEasy
- 69Computing 3^123bmod100 by repeated squaringAlgebraFundamental Theorem of Arithmetic, Congruences, Chinese Remainder TheoremEasy
- 70Fermat's two-square theorem flags the prime 11equiv3pmod4 to an odd powerAlgebraFundamental Theorem of Arithmetic, Congruences, Chinese Remainder TheoremEasy
- 71Non-holonomic constraints leave the configuration-space dimension unreducedClassical MechanicsGeneralized Coordinates, Lagrange's EquationsEasy
- 72The momentum conjugate to a cyclic coordinate is conservedClassical MechanicsGeneralized Coordinates, Lagrange's EquationsEasy
- 73Hamilton's equations for H=pq give exponential growth and decayClassical MechanicsHamilton's Canonical Equations, Hamilton's PrincipleEasy
- 74Equal transverse moments of inertia make the spin-axis angular velocity constantClassical MechanicsMotion of Rigid Bodies, Euler's Dynamical Equations, Small OscillationsEasy
- 75Irrotationality is unrelated to mass conservationFluid DynamicsPotential Flow, Streamlines, DoubletsMedium
- 76Recovering the potential with the q=-nablaphi sign conventionFluid DynamicsPotential Flow, Streamlines, DoubletsEasy
- 77Both potential functions, under the physics sign conventionFluid DynamicsPotential Flow, Streamlines, DoubletsMedium
- 78The image of a doublet in a straight-line boundaryFluid DynamicsPotential Flow, Streamlines, DoubletsMedium
- 79Gaussian curvature as the ratio of the two fundamental form determinantsLinear AlgebraAlgebra of Matrices, Rank, Determinant, Linear EquationsMedium
- 80The Frenet formula bar t'=kappabar nLinear AlgebraInner Product Spaces, Orthonormal BasisEasy
- 81A hyperbolic paraboloid has strictly negative Gaussian curvature everywhereDifferential GeometryCurves and Surfaces, CurvatureEasy
- 82A resonant boundary condition leaves one constant completely freeCalculus of VariationsVariation of a Functional, Euler-Lagrange Equation, Conditions for ExtremaEasy
- 83The Euler equation for a Dirichlet-type surface functional is Poisson's equationPartial Differential EquationsSeparation of Variables (Laplace, Heat and Wave Equations)Easy
- 84A fourth-order Euler–Poisson equation solved exactly by sin xOrdinary Differential EquationsGeneral Theory of Linear ODEs, Variation of Parameters, Sturm-Liouville, Green's FunctionMedium
- 85The geodesic on a sphereCalculus of VariationsVariation of a Functional, Euler-Lagrange Equation, Conditions for ExtremaEasy
- 86The standard IVP/BVP integral-equation reductions are exactly reversed hereAnalysisSequences and SeriesMedium
- 87Building up the iterated kernel for k(t,s)=e^t-sLinear Integral EquationsFredholm and Volterra Type, Separable Kernels, Resolvent KernelEasy
- 88Solving a convolution-kernel Volterra equation via Laplace transformsLinear Integral EquationsFredholm and Volterra Type, Separable Kernels, Resolvent KernelEasy
- 89delta is a difference E^1/2-E^-1/2, not a sumNumerical AnalysisFinite Differences, Lagrange/Hermite/Spline InterpolationMedium
- 90One step of Newton-Raphson from x0=0Numerical AnalysisNumerical Solutions of Algebraic Equations, Newton-Raphson, Rate of ConvergenceEasy
- 91One step of Euler's methodOrdinary Differential EquationsExistence & Uniqueness, Singular Solutions, Systems of First Order ODEsEasy
- 92The second shifting theorem for Laplace transformsIntegral TransformsLaplace TransformEasy
- 93The Fourier sine transform of e^-at/tIntegral TransformsFourier TransformEasy
- 94Laplace transform of a repeated-root second-order IVPOrdinary Differential EquationsGeneral Theory of Linear ODEs, Variation of Parameters, Sturm-Liouville, Green's FunctionEasy
- 95Balanced supply and demand is both necessary and sufficient for transportation feasibilityOperations ResearchTransportation and Assignment ProblemsMedium
- 96Excluding the pentagon N5 from the five lattices on 5 elementsAlgebraLattice Theory (Distributive/Modular/Boolean Lattices)Easy
- 97Neither distributivity nor modularity alone guarantees complementation existsAlgebraLattice Theory (Distributive/Modular/Boolean Lattices)Medium
- 98sin x's integral over an unbounded set is not even well-definedAnalysisMonotonic Functions, Bounded Variation, Lebesgue Measure & IntegralMedium
- 99Disjoint supports make two measures mutually singularAnalysisMonotonic Functions, Bounded Variation, Lebesgue Measure & IntegralMedium
- 100Lebesgue integrability is equivalent to absolute integrabilityAnalysisRiemann Sums and Riemann Integral, Improper IntegralsEasy
Section III · Statistics · Q101–190
Practise this section →- 101Identifying which side of the percentile boundary the shift applies toDescriptive StatisticsDescriptive Statistics, Exploratory Data AnalysisEasy
- 102Covariance is bilinear: shifts vanish, scalars multiply outProbabilityRandom Variables and Distribution Functions, Expectation and MomentsEasy
- 103A common variance means quantile ordering is decided purely by the meansProbabilityRandom Variables and Distribution Functions, Expectation and MomentsMedium
- 104P(Amid B)=P(B) forces P(B)^2=P(Acap B)le P(A)ProbabilitySample Space, Discrete Probability, Independent Events, Bayes TheoremEasy
- 105Bounding E[X] by the range of the free probability massProbabilityRandom Variables and Distribution Functions, Expectation and MomentsEasy
- 106Independence lets a compound conditioning event drop outProbabilityRandom Variables and Distribution Functions, Expectation and MomentsEasy
- 107MGF of a sum multiplies; MGF of a product needs conditioningProbabilityRandom Variables and Distribution Functions, Expectation and MomentsMedium
- 108E[YX]=X^2 for standard normal X forces zero correlation, not positiveProbabilityRandom Variables and Distribution Functions, Expectation and MomentsEasy
- 109The minimum of n iid exponentials is exponential with mean theta/nDistribution TheoryStandard Discrete and Continuous Univariate DistributionsEasy
- 110Squaring a standard normal quantile gives the chi-square quantileStatistical InferenceAnalysis of Discrete Data and Chi-Square Test of Goodness of FitEasy
- 111A mixed discrete-continuous joint law, checked term by termProbabilityRandom Variables and Distribution Functions, Expectation and MomentsHard
- 112The Mahalanobis quadratic form is always chi-square distributedMultivariate AnalysisMultivariate Normal Distribution, Wishart Distribution, Quadratic FormsEasy
- 113The MLE on a restricted parameter space is a boundary-projected sample meanStatistical InferenceMethods of Estimation, Properties of EstimatorsMedium
- 114Combining a lower and upper confidence limit into a two-sided intervalStatistical InferenceConfidence IntervalsEasy
- 115Computing test size directly under the discrete nullStatistical InferenceTests of Hypotheses: MP and UMP Tests, Likelihood Ratio TestsEasy
- 116The Rao–Blackwell–Lehmann–Scheffe theorem constructs the UMVUEStatistical InferenceMethods of Estimation, Properties of EstimatorsEasy
- 117Likelihood decreasing in theta picks the smallest feasible value, X(n)/5Statistical InferenceMethods of Estimation, Properties of EstimatorsEasy
- 118The chi^2n-1 null distribution for the sample-variance test statistic requires normalityStatistical InferenceTests of Hypotheses: MP and UMP Tests, Likelihood Ratio TestsMedium
- 119A larger population variance widens the confidence interval proportionallyStatistical InferenceConfidence IntervalsEasy
- 120A multiple correlation coefficient can never be negativeMultivariate AnalysisPartial and Multiple Correlation CoefficientsEasy
- 121Chi-square degrees of freedom: one for the total, one per estimated parameterStatistical InferenceAnalysis of Discrete Data and Chi-Square Test of Goodness of FitEasy
- 122Kendall's tau from the concordant count: (C-D)/binom n2Statistical InferenceNonparametric Tests, Rank Correlation, Test for IndependenceEasy
- 123Group replacement of bulbs: the printed costs make one week optimalOperations ResearchReplacement ModelsHard
- 124The EOQ is where ordering and carrying costs balanceOperations ResearchElementary Queuing and Inventory ModelsEasy
- 125Squaring the traffic intensity lowers it, so the second queue is shorterOperations ResearchSteady-State Solutions of Markovian Queuing Models (M/M/1, M/M/C, M/G/1)Medium
- 126Random halving, then one unit per half, gives every pair the same chanceSampling TheorySimple Random, Stratified and Systematic SamplingMedium
- 127Proportional allocation never does worse than SRSWOR of the same sizeSampling TheorySimple Random, Stratified and Systematic SamplingEasy
- 128An F-ratio needs no degrees-of-freedom correction when the two d.f. are equalDesign of ExperimentsCompletely Randomized, Randomized Block and Latin-Square DesignsEasy
- 129In one-way ANOVA, cmu+sum dialphai is estimable iff sum di=cLinear Models & RegressionGauss-Markov Models, Estimability, Best Linear Unbiased EstimatorsMedium
- 130A Vtimes V Latin square uses V^2 plots, not V^3Design of ExperimentsCompletely Randomized, Randomized Block and Latin-Square DesignsEasy
- 131Finite additivity alone gives superadditivity over countable disjoint unionsProbabilityRandom Variables and Distribution Functions, Expectation and MomentsMedium
- 132A bounded sequence is not dominated by an integrable function on an infinite measure spaceProbabilityRandom Variables and Distribution Functions, Expectation and MomentsMedium
- 133A measurable map must be constant on every atom of the sigma-fieldProbabilityRandom Variables and Distribution Functions, Expectation and MomentsMedium
- 134A bounded function of i.i.d. variables always obeys the SLLNProbabilityModes of Convergence, Laws of Large Numbers, Central Limit TheoremMedium
- 135CLT for the squares: a finite fourth moment gives a normal limitProbabilityModes of Convergence, Laws of Large Numbers, Central Limit TheoremEasy
- 136Here Y is uniform, so Xmid Y=1 is a unit exponentialProbabilityRandom Variables and Distribution Functions, Expectation and MomentsMedium
- 137Almost-sure convergence does not imply mean-square convergenceProbabilityModes of Convergence, Laws of Large Numbers, Central Limit TheoremEasy
- 138As printed the double intersection is not a tail event; read as liminf An it isProbabilitySample Space, Discrete Probability, Independent Events, Bayes TheoremHard
- 139A square-integrable martingale's second moment is non-decreasing, not constantStochastic ProcessesMartingalesMedium
- 140A distribution function with two jumps and a uniform stretchProbabilityRandom Variables and Distribution Functions, Expectation and MomentsMedium
- 141A mixed law: uniform density tfrac12 on [0,1) plus an atom of tfrac12 at 1ProbabilityRandom Variables and Distribution Functions, Expectation and MomentsEasy
- 142Markov's inequality applied to g(X)ProbabilityCharacteristic Functions, Probability InequalitiesEasy
- 143The Cauchy characteristic function: e^itmu-sigmatProbabilityCharacteristic Functions, Probability InequalitiesEasy
- 144Poisson: binomial given the total, shifted Poisson given one summandDistribution TheoryStandard Discrete and Continuous Univariate DistributionsEasy
- 145Minimax means the smallest worst-case risk: d2Statistical InferenceElementary Bayesian InferenceEasy
- 146The gamma family is the conjugate prior for a Poisson meanStatistical InferenceElementary Bayesian InferenceEasy
- 147The joint density depends on the data only through X1+2X2Statistical InferenceMethods of Estimation, Properties of EstimatorsMedium
- 148A ratio of unbiased estimators is not unbiasedStatistical InferenceMethods of Estimation, Properties of EstimatorsEasy
- 149For U(theta,theta+1) any point of [X(n)-1,X(1)] is an MLEStatistical InferenceMethods of Estimation, Properties of EstimatorsMedium
- 150A randomised Neyman–Pearson test: reject at x=4 with probability 0.2Statistical InferenceTests of Hypotheses: MP and UMP Tests, Likelihood Ratio TestsMedium
- 151Sn^2 is unbiased for sigma^2, but Sn is biased for sigmaStatistical InferenceMethods of Estimation, Properties of EstimatorsEasy
- 152Dividing by sum Xi^2 rather than its square root leaves a N(0,1/sigma^2) limitDistribution TheorySampling Distributions, Standard Errors, Asymptotic DistributionsMedium
- 153A U-statistic is the Rao–Blackwellisation of any unbiased kernel estimatorStatistical InferenceNonparametric Tests, Rank Correlation, Test for IndependenceMedium
- 154The ARE of the zero-frequency estimator is lambda/(e^lambda-1), not a listed optionDistribution TheorySampling Distributions, Standard Errors, Asymptotic DistributionsHard
- 155A vanishing c/sqrt n shift keeps consistency but moves the normal limit to N(c,1)Distribution TheorySampling Distributions, Standard Errors, Asymptotic DistributionsMedium
- 156Equicorrelated normals: sums of two coordinates share one lawMultivariate AnalysisMultivariate Normal Distribution, Wishart Distribution, Quadratic FormsMedium
- 157Flipping the sign on X1le3 keeps the N(0,1) marginal but makes the pair strongly negatively correlatedMultivariate AnalysisMultivariate Normal Distribution, Wishart Distribution, Quadratic FormsMedium
- 158X1+X3 and X1-X2 are proportional, so their joint law is singularMultivariate AnalysisMultivariate Normal Distribution, Wishart Distribution, Quadratic FormsEasy
- 159S11+S12 can be negative, so it is never a multiple of a chi-squareMultivariate AnalysisMultivariate Normal Distribution, Wishart Distribution, Quadratic FormsHard
- 160Mixed moments from the c.f.: divide the derivative at underline0 by i^4=1Multivariate AnalysisMultivariate Normal Distribution, Wishart Distribution, Quadratic FormsEasy
- 161Estimable functions are the combinations of theta2-theta1 and theta2+theta3Linear Models & RegressionGauss-Markov Models, Estimability, Best Linear Unbiased EstimatorsEasy
- 162Residuals sum to zero whenever the model has an interceptLinear Models & RegressionSimple and Multiple Linear Regression, Logistic RegressionEasy
- 163A fitted regression predicts; it neither proves causation nor needs a good fit to existLinear Models & RegressionSimple and Multiple Linear Regression, Logistic RegressionEasy
- 164Only the error mean square is unbiased for sigma^2 in generalLinear Models & RegressionSimple and Multiple Linear Regression, Logistic RegressionHard
- 165Centred regressor values make the least-squares estimators uncorrelatedLinear Models & RegressionGauss-Markov Models, Estimability, Best Linear Unbiased EstimatorsEasy
- 166Inclusion probability under successive PPS draws: z3+sumine3ziz3/(1-zi)Sampling TheoryProbability Proportional to Size SamplingMedium
- 167Bias of the ratio estimator from Cov(r,bar x)=bar Y-bar X,E(r)Sampling TheoryRatio and Regression MethodsMedium
- 168Regression beats both, but mean versus ratio depends on rhoSampling TheoryRatio and Regression MethodsMedium
- 169A block holding a single treatment isolates it: disconnected and non-orthogonalDesign of ExperimentsConnectedness and Orthogonality of Block Designs, BIBDMedium
- 170For a BIBD, NN'=(r-lambda)I+lambda J is non-singularDesign of ExperimentsConnectedness and Orthogonality of Block Designs, BIBDEasy
- 171Eight treatments, twice each, in a 4times4 square: a quasi-Latin squareDesign of Experiments2^K Factorial Experiments: Confounding and ConstructionMedium
- 172As printed, the blocks split on the parity of b,d — BD, not a listed effectDesign of Experiments2^K Factorial Experiments: Confounding and ConstructionMedium
- 173The random-phase sinusoid has autocovariance tfrac12cos(omega h)ProbabilityRandom Variables and Distribution Functions, Expectation and MomentsMedium
- 174A random walk's variance grows linearly, so only Xt/sqrt t has stable varianceProbabilityRandom Variables and Distribution Functions, Expectation and MomentsMedium
- 175Variance of a mean of three AR(1) values needs autocovariances, not autocorrelationsProbabilityRandom Variables and Distribution Functions, Expectation and MomentsHard
- 176An irreducible chain of period 3: a unique stationary law, but no limitStochastic ProcessesMarkov Chains: Classification of States, Stationary DistributionMedium
- 177min(s,t)/sqrtst is the Poisson process correlation; the covariance is lambdamin(s,t)Stochastic ProcessesPoisson and Birth-and-Death ProcessesEasy
- 178A symmetric walk reflected at 0 is null recurrentStochastic ProcessesMarkov Chains: Classification of States, Stationary DistributionMedium
- 179A finite expected total progeny forces m<1 and certain extinctionStochastic ProcessesPoisson and Birth-and-Death ProcessesMedium
- 180Standardisation removes the effect of differing population compositionDemography / Vital StatisticsLife Tables, Mortality RateEasy
- 181Linear, exponential and logistic are standard; BC^t reads as a garbled Gompertz curveDemography / Vital StatisticsLife Tables, Mortality RateMedium
- 182Exponential reliability at 800 hours rises from a^2 to aSurvival AnalysisHazard Function and Failure Rates, Censoring and Life TestingEasy
- 183Integrating the hazard: H(t)=t-ln(1+t) gives S(t)=(1+t)e^-tSurvival AnalysisHazard Function and Failure Rates, Censoring and Life TestingEasy
- 184Reorder level = usage during the lead timeOperations ResearchElementary Queuing and Inventory ModelsEasy
- 185Stock S=Q-ut hits zero at t=Q/uOperations ResearchElementary Queuing and Inventory ModelsEasy
- 186The M/M/K departure rate caps at kmu — no printed option says soOperations ResearchSteady-State Solutions of Markovian Queuing Models (M/M/1, M/M/C, M/G/1)Easy
- 187Two stages split c evenly: f2(5)=25/4 (the key's 125/27 is f3(5))Operations ResearchLinear Programming Problem, Simplex Method, DualityMedium
- 188Equal parts maximise a product with a fixed sumOperations ResearchDynamic and Integer ProgrammingEasy
- 189Dual prices are CB'B^-1, and the dual simplex need not end degenerateOperations ResearchLinear Programming Problem, Simplex Method, DualityMedium
- 190The DP stage for x2 carries both constraints that involve x2Operations ResearchLinear Programming Problem, Simplex Method, DualityMedium