March 2023
180 questions, booklet MAR-30223.
Section I · General (compulsory) · Q1–20
Practise this section →- 1Two disguised descriptions of the same set of odd integersAnalysisElementary Set Theory, Real Number System, Supremum, InfimumEasy
- 2The alternating harmonic seriesComplex AnalysisTaylor Series, Laurent Series, Calculus of ResiduesEasy
- 3A principal-value Mellin-type integralAnalysisRiemann Sums and Riemann Integral, Improper IntegralsMedium
- 4The agreement set of two continuous maps is always closedTopologySeparation Axioms, Connectedness and CompactnessEasy
- 5Spotting the pair with genuinely different cardinalitiesAnalysisElementary Set Theory, Real Number System, Supremum, InfimumMedium
- 6Set limits — every point misses infinitely many termsAnalysisSequences and SeriesMedium
- 7The countable-or-co-countable sigma-field — an interval that belongs to neitherAnalysisElementary Set Theory, Real Number System, Supremum, InfimumHard
- 8Matching value and slope at a piecewise junctionAnalysisContinuity, Uniform Continuity, Differentiability, Mean Value TheoremEasy
- 9Filling a removable discontinuity via a half-angle identityAnalysisContinuity, Uniform Continuity, Differentiability, Mean Value TheoremEasy
- 10Partial derivatives fail to exist for this classic non-differentiable exampleAnalysisSequences and SeriesMedium
- 11Checking three sets for linear independence — and finding all three passLinear AlgebraVector Spaces, Subspaces, Linear Dependence, Basis, DimensionHard
- 12Dimension of the singular locus inside the symmetric matricesLinear AlgebraVector Spaces, Subspaces, Linear Dependence, Basis, DimensionEasy
- 13Recovering T(3,4) from linearity and two known valuesLinear AlgebraAlgebra of Linear TransformationsEasy
- 14AB=0 always means the column space of B sits inside the null space of ALinear AlgebraAlgebra of Linear TransformationsHard
- 15Shifting by a scalar multiple of I preserves eigenvectorsLinear AlgebraAlgebra of Matrices, Rank, Determinant, Linear EquationsMedium
- 16Determinant of a function of a matrix, via its eigenvaluesLinear AlgebraAlgebra of Matrices, Rank, Determinant, Linear EquationsEasy
- 17What matching characteristic polynomials guarantee — and what they don'tLinear AlgebraAlgebra of Matrices, Rank, Determinant, Linear EquationsMedium
- 18Several operators fail to be nilpotent — for overlapping reasonsAnalysisMetric Spaces, Compactness, Connectedness, Normed Linear SpacesHard
- 19The one orthogonal-complement fact that needs no extra hypothesesLinear AlgebraInner Product Spaces, Orthonormal BasisMedium
- 20Symmetric matrices are always diagonalizable — collapsing the Jordan form to the identityLinear AlgebraMatrix Representation, Change of Basis, Canonical and Jordan FormsMedium
Section II · Mathematics · Q21–100
Practise this section →- 21Evaluating a complex power directly from the principal logarithmComplex AnalysisAlgebra of Complex Numbers, Power Series, Transcendental FunctionsEasy
- 22Recovering u from v via the Cauchy–Riemann equationsComplex AnalysisAnalytic Functions, Cauchy-Riemann EquationsEasy
- 23Radius of convergence of a geometric-type power seriesComplex AnalysisAlgebra of Complex Numbers, Power Series, Transcendental FunctionsEasy
- 24The family of disk automorphisms sending 1/2 to 0Complex AnalysisAlgebra of Complex Numbers, Power Series, Transcendental FunctionsMedium
- 25A Möbius map with infty as its only fixed point is a translationComplex AnalysisConformal Mappings, Mobius TransformationsEasy
- 26Open mapping theorem — when the image is forced to be a single pointAnalysisMetric Spaces, Compactness, Connectedness, Normed Linear SpacesMedium
- 27Image of a square under the exponential mapComplex AnalysisAlgebra of Complex Numbers, Power Series, Transcendental FunctionsEasy
- 28Residue extraction from the Laurent series of z^4sin(1/z)Complex AnalysisTaylor Series, Laurent Series, Calculus of ResiduesEasy
- 29Radius of convergence via the ratio test and eAnalysisSequences and SeriesEasy
- 30Weierstrass's theorem — uniform limits of analytic functions are analyticAnalysisSequences and SeriesEasy
- 31Recovering f from a convergent sequence of values inside a diskAnalysisSequences and SeriesEasy
- 32Two theorems in tandem — a bounded-real-part trick and equality in Schwarz–PickComplex AnalysisLiouville's Theorem, Maximum Modulus Principle, Open Mapping TheoremHard
- 33Residue of cot z at a pole on the real axisComplex AnalysisTaylor Series, Laurent Series, Calculus of ResiduesEasy
- 34Casorati–Weierstrass — the image near an essential singularity is denseComplex AnalysisLiouville's Theorem, Maximum Modulus Principle, Open Mapping TheoremMedium
- 35Degree of the splitting field of x^3-2Complex AnalysisAlgebra of Complex Numbers, Power Series, Transcendental FunctionsEasy
- 36Finite extensions of mathbf R — only mathbf R and mathbf C existAnalysisSequences and SeriesEasy
- 37Spotting the false claim among four algebra factsAlgebraFields, Finite Fields, Field Extensions, Galois TheoryMedium
- 38Counting low-degree irreducible polynomials over mathbf Z2AlgebraFields, Finite Fields, Field Extensions, Galois TheoryEasy
- 39Maximal principal ideals of F[x] are generated by irreduciblesAlgebraRings, Ideals, UFD, PID, Euclidean DomainEasy
- 40Spotting the false claim about Euclidean domains and PIDsAlgebraRings, Ideals, UFD, PID, Euclidean DomainMedium
- 41mathbf Z[x] is neither a PID nor EuclideanAlgebraRings, Ideals, UFD, PID, Euclidean DomainEasy
- 42Which power maps are ring homomorphisms — checking with a concrete counterexampleAlgebraGroups, Subgroups, Sylow Theorems, Cayley's TheoremMedium
- 43Finite order of every element does not force a finite groupAlgebraGroups, Subgroups, Sylow Theorems, Cayley's TheoremMedium
- 44p-groups always have a non-trivial centerAlgebraGroups, Subgroups, Sylow Theorems, Cayley's TheoremHard
- 45An abelian group always has a subgroup of order lcm(m,n)AlgebraGroups, Subgroups, Sylow Theorems, Cayley's TheoremMedium
- 46Inn(S3)cong S3, not trivialAlgebraGroups, Subgroups, Sylow Theorems, Cayley's TheoremMedium
- 47Counting distinct monomial terms in a multinomial expansionAlgebraPermutations, Combinations, Pigeon-Hole Principle, Inclusion-ExclusionEasy
- 48Last digit of 3^55 via the period-4 cycleAlgebraFundamental Theorem of Arithmetic, Congruences, Chinese Remainder TheoremEasy
- 49A linear map from R^2 compresses a compact square into a connected imageLinear AlgebraAlgebra of Linear TransformationsMedium
- 50Set difference of a half-open rectangle and a disk — when it becomes compactAnalysisMetric Spaces, Compactness, Connectedness, Normed Linear SpacesHard
- 51Why R and R^2 cannot be homeomorphicTopologyBasis, Dense Sets, Subspace and Product TopologyHard
- 52Unions of dense sets are always dense, unlike intersections or differencesAnalysisMetric Spaces, Compactness, Connectedness, Normed Linear SpacesMedium
- 53Convex sets are always connected — and a bowtie union is not even convexAnalysisElementary Set Theory, Real Number System, Supremum, InfimumMedium
- 54Density is preserved by continuous surjectionsTopologyBasis, Dense Sets, Subspace and Product TopologyEasy
- 55Spotting the collection that can't isolate a single integer sliceTopologyBasis, Dense Sets, Subspace and Product TopologyHard
- 56A K-topology-style refinement — Hausdorff, but not regularTopologyBasis, Dense Sets, Subspace and Product TopologyMedium
- 57Neither ell^p nor ell^q is a closed subspace of the otherAnalysisMetric Spaces, Compactness, Connectedness, Normed Linear SpacesMedium
- 58Both candidate distances fail identity of indiscerniblesAnalysisMetric Spaces, Compactness, Connectedness, Normed Linear SpacesMedium
- 59Removing a discrete lattice or a parabola from the planeAnalysisElementary Set Theory, Real Number System, Supremum, InfimumHard
- 60Gluing the endpoints of an interval produces a circleLinear AlgebraVector Spaces, Subspaces, Linear Dependence, Basis, DimensionEasy
- 61Lipschitz continuity, not mere continuity or boundedness, guarantees uniquenessAnalysisContinuity, Uniform Continuity, Differentiability, Mean Value TheoremMedium
- 62A shared zero forces the Wronskian to vanish identicallyOrdinary Differential EquationsGeneral Theory of Linear ODEs, Variation of Parameters, Sturm-Liouville, Green's FunctionMedium
- 63Differential equation of a family of circles centred on the x-axisOrdinary Differential EquationsGeneral Theory of Linear ODEs, Variation of Parameters, Sturm-Liouville, Green's FunctionEasy
- 64Two functions with nonvanishing Wronskian can always be realised as ODE solutionsLinear AlgebraVector Spaces, Subspaces, Linear Dependence, Basis, DimensionMedium
- 65Picard's existence interval, optimised over the rectangle's heightOrdinary Differential EquationsExistence & Uniqueness, Singular Solutions, Systems of First Order ODEsMedium
- 66Solving a piecewise linear ODE with x forcingAnalysisContinuity, Uniform Continuity, Differentiability, Mean Value TheoremMedium
- 67Orthogonal trajectories of a parabola familyOrdinary Differential EquationsGeneral Theory of Linear ODEs, Variation of Parameters, Sturm-Liouville, Green's FunctionEasy
- 68The characteristics of yp-xq=0 are exactly the shared intersection circlesPartial Differential EquationsLagrange and Charpit Methods, Cauchy Problem for First Order PDEsEasy
- 69Cauchy problems for a PDE can have no, one, or infinitely many surfacesOrdinary Differential EquationsExistence & Uniqueness, Singular Solutions, Systems of First Order ODEsMedium
- 70A transversal curve gives a unique paraboloid of revolutionPartial Differential EquationsLagrange and Charpit Methods, Cauchy Problem for First Order PDEsMedium
- 71The compatibility condition for two first-order PDEsAnalysisFunctions of Several Variables, Inverse & Implicit Function TheoremsMedium
- 72Integral surfaces and characteristic curves — a two-way theoremPartial Differential EquationsLagrange and Charpit Methods, Cauchy Problem for First Order PDEsMedium
- 73Compatibility means a genuinely shared solutionPartial Differential EquationsSeparation of Variables (Laplace, Heat and Wave Equations)Easy
- 74Pfaffian and Lagrange normals describe the same surfacesPartial Differential EquationsLagrange and Charpit Methods, Cauchy Problem for First Order PDEsMedium
- 75Classifying a variable-coefficient PDE via its non-negative discriminantPartial Differential EquationsClassification of Second Order PDEsEasy
- 76Finite differences of the factorial polynomialNumerical AnalysisFinite Differences, Lagrange/Hermite/Spline InterpolationMedium
- 77nth forward difference of the exponential functionNumerical AnalysisFinite Differences, Lagrange/Hermite/Spline InterpolationEasy
- 78Deriving the Newton–Raphson iteration for x=e^-xNumerical AnalysisNumerical Solutions of Algebraic Equations, Newton-Raphson, Rate of ConvergenceEasy
- 79Equivalent central-difference interpolation formulasNumerical AnalysisFinite Differences, Lagrange/Hermite/Spline InterpolationEasy
- 80Fifth difference of a degree-5 polynomialNumerical AnalysisFinite Differences, Lagrange/Hermite/Spline InterpolationEasy
- 81Finding a missing table entry by forcing the top difference to vanishNumerical AnalysisFinite Differences, Lagrange/Hermite/Spline InterpolationEasy
- 82The brachistochrone's defining first-order ODEOrdinary Differential EquationsGeneral Theory of Linear ODEs, Variation of Parameters, Sturm-Liouville, Green's FunctionEasy
- 83The isoperimetric theorem — the circle maximises area for fixed perimeterCalculus of VariationsVariation of a Functional, Euler-Lagrange Equation, Conditions for ExtremaEasy
- 84An extremal problem reducing to 4y=x(x-1)Calculus of VariationsVariation of a Functional, Euler-Lagrange Equation, Conditions for ExtremaEasy
- 85Euler equations for a two-variable parametric functional, combined into a 4th-order ODEOrdinary Differential EquationsGeneral Theory of Linear ODEs, Variation of Parameters, Sturm-Liouville, Green's FunctionMedium
- 86The fully expanded Euler–Lagrange equationCalculus of VariationsVariation of a Functional, Euler-Lagrange Equation, Conditions for ExtremaMedium
- 87The variational form of Poisson's equationTopologyBasis, Dense Sets, Subspace and Product TopologyMedium
- 88A separable-kernel eigenfunction shapeLinear Integral EquationsFredholm and Volterra Type, Separable Kernels, Resolvent KernelEasy
- 89Recognising a separable (degenerate) kernelLinear AlgebraAlgebra of Linear TransformationsEasy
- 90Converting x"-x=0, x(0)=1,x'(0)=1 into a Volterra equationOrdinary Differential EquationsExistence & Uniqueness, Singular Solutions, Systems of First Order ODEsEasy
- 91Symmetric kernels give real eigenvalues and orthogonal eigenfunctionsAnalysisMetric Spaces, Compactness, Connectedness, Normed Linear SpacesMedium
- 92Solving a Volterra equation via differentiationAnalysisContinuity, Uniform Continuity, Differentiability, Mean Value TheoremEasy
- 93Eigenvalues of a degenerate (separable) Fredholm kernelLinear Integral EquationsFredholm and Volterra Type, Separable Kernels, Resolvent KernelMedium
- 94Total energy in Hamiltonian (momentum) form for projectile motionClassical MechanicsHamilton's Canonical Equations, Hamilton's PrincipleEasy
- 95Hamiltonian for a velocity-dependent (non-natural) potentialClassical MechanicsHamilton's Canonical Equations, Hamilton's PrincipleHard
- 96H is conserved even when it fails to equal the total energyClassical MechanicsHamilton's Canonical Equations, Hamilton's PrincipleHard
- 97Speed-squared after time t for oblique projectile motionClassical MechanicsMotion of Rigid Bodies, Euler's Dynamical Equations, Small OscillationsEasy
- 98Identifying the Hamiltonian for a reduced (effective) one-coordinate LagrangianClassical MechanicsGeneralized Coordinates, Lagrange's EquationsHard
- 99Lagrangian for a body falling under gravity (downward-positive coordinate)Classical MechanicsGeneralized Coordinates, Lagrange's EquationsEasy
- 100Degrees of freedom of a rigid body with one point fixedClassical MechanicsMotion of Rigid Bodies, Euler's Dynamical Equations, Small OscillationsEasy
Section III · Statistics · Q101–180
Practise this section →- 101Recognising a binomial PGFProbabilityRandom Variables and Distribution Functions, Expectation and MomentsEasy
- 102Normal distribution facts — mean deviation coincides at a symmetric centreDistribution TheoryStandard Discrete and Continuous Univariate DistributionsMedium
- 103Mode of grouped data from the modal-class formulaDescriptive StatisticsDescriptive Statistics, Exploratory Data AnalysisEasy
- 104A simple counting-principle questionAlgebraPermutations, Combinations, Pigeon-Hole Principle, Inclusion-ExclusionEasy
- 105Four probability facts — only subadditivity survives unconditionallyProbabilityRandom Variables and Distribution Functions, Expectation and MomentsHard
- 106Reading probability off a piecewise CDF, including a jumpProbabilityRandom Variables and Distribution Functions, Expectation and MomentsMedium
- 107A false moment inequality — checked by direct counterexampleProbabilityRandom Variables and Distribution Functions, Expectation and MomentsMedium
- 108Two indicator variables that are neither identically distributed nor independentProbabilityRandom Variables and Distribution Functions, Expectation and MomentsMedium
- 109Reading facts off an empirical distribution from raw dataProbabilityRandom Variables and Distribution Functions, Expectation and MomentsMedium
- 110A four-point discrete distribution built from degenerate mixturesProbabilityRandom Variables and Distribution Functions, Expectation and MomentsMedium
- 111Law of total variance for a mixed Binomial–Geometric hierarchyProbabilityRandom Variables and Distribution Functions, Expectation and MomentsMedium
- 112Jensen's inequality applied to four power transformations — x^1/4 is concave, not convexProbabilityCharacteristic Functions, Probability InequalitiesEasy
- 113Recognising e^-t as the standard Cauchy (= t1) characteristic functionProbabilityCharacteristic Functions, Probability InequalitiesEasy
- 114Borel–Cantelli I and its contrapositive, without needing independenceProbabilityModes of Convergence, Laws of Large Numbers, Central Limit TheoremHard
- 115Extreme order statistics converge to a point, not to a rescaled limit lawDistribution TheoryStandard Discrete and Continuous Univariate DistributionsHard
- 116Multiplying a fixed random variable by an indicator converging to 1ProbabilityRandom Variables and Distribution Functions, Expectation and MomentsMedium
- 117Slutsky's theorem — convergence in distribution is all that's guaranteedProbabilityRandom Variables and Distribution Functions, Expectation and MomentsHard
- 118Rates of growth for sums and sums-of-squares of iid mean-zero unit-variance variablesProbabilityModes of Convergence, Laws of Large Numbers, Central Limit TheoremHard
- 119Mean recurrence time versus first-return probabilityStochastic ProcessesMarkov Chains: Classification of States, Stationary DistributionMedium
- 120First-visit probabilities into a closed class — when they're forced to be 1, and when they're notStochastic ProcessesMarkov Chains: Classification of States, Stationary DistributionHard
- 121First-return probability is bounded by return probability; recurrence forces divergenceStochastic ProcessesMarkov Chains: Classification of States, Stationary DistributionMedium
- 122Non-null persistence (not transience, null persistence, or mere ergodicity) is the exact requirementStochastic ProcessesMarkov Chains: Classification of States, Stationary DistributionMedium
- 123Waiting past two weeks with an exponential lifetimeDistribution TheoryStandard Discrete and Continuous Univariate DistributionsEasy
- 124A pure death process — one absorbing state, everything else transientStochastic ProcessesMarkov Chains: Classification of States, Stationary DistributionMedium
- 125A Poisson approximation for rare disease counts in a large new sampleProbabilityRandom Variables and Distribution Functions, Expectation and MomentsMedium
- 126Expected maximum of two iid standard normalsProbabilityRandom Variables and Distribution Functions, Expectation and MomentsEasy
- 127Finding the PGF that gives variance exactly 1ProbabilityRandom Variables and Distribution Functions, Expectation and MomentsMedium
- 128Multinomial marginals are binomial — but never independent of each otherDistribution TheoryStandard Discrete and Continuous Univariate DistributionsHard
- 129A symmetry argument for the conditional mean of an exponential splitProbabilityRandom Variables and Distribution Functions, Expectation and MomentsMedium
- 130Cochran's theorem for correlated normals — only the whitened form is chi-squareMultivariate AnalysisMultivariate Normal Distribution, Wishart Distribution, Quadratic FormsHard
- 131MLE of a Bernoulli parameter restricted to [1/3,2/3]Statistical InferenceMethods of Estimation, Properties of EstimatorsEasy
- 132Unbiased estimation of p^3 via a falling-factorial momentStatistical InferenceMethods of Estimation, Properties of EstimatorsEasy
- 133Size of a test with a critical region unreachable under H0Statistical InferenceTests of Hypotheses: MP and UMP Tests, Likelihood Ratio TestsEasy
- 134UMVUE of e^-theta for Poisson data — a binomial thinning argumentStatistical InferenceMethods of Estimation, Properties of EstimatorsMedium
- 135Minimal sufficiency survives under a bijective relabellingProbabilityRandom Variables and Distribution Functions, Expectation and MomentsHard
- 136Setting up a one-sided test for underperformanceDesign of Experiments2^K Factorial Experiments: Confounding and ConstructionMedium
- 137UMVUE of theta^2 for a normal mean with known varianceStatistical InferenceMethods of Estimation, Properties of EstimatorsEasy
- 138Spotting the irregular family — support depending on the parameterStatistical InferenceMethods of Estimation, Properties of EstimatorsHard
- 139A consistent moment-based estimator for the Laplace scaleStatistical InferenceMethods of Estimation, Properties of EstimatorsHard
- 140Asymptotic variance of the sample median for normal dataProbabilityRandom Variables and Distribution Functions, Expectation and MomentsMedium
- 141The sample mean of Cauchy data never concentratesProbabilityModes of Convergence, Laws of Large Numbers, Central Limit TheoremHard
- 142The combinatorial definition of the Mann–Whitney U statisticProbabilityRandom Variables and Distribution Functions, Expectation and MomentsHard
- 143Two standard Fisher information constantsProbabilityRandom Variables and Distribution Functions, Expectation and MomentsMedium
- 144Matched-pair design calls for the paired t-testProbabilityRandom Variables and Distribution Functions, Expectation and MomentsMedium
- 145A posterior formula that reproduces both point masses of a two-point priorStatistical InferenceElementary Bayesian InferenceHard
- 146Poisson–Gamma posterior predictive is Negative BinomialStatistical InferenceElementary Bayesian InferenceHard
- 147Normality of errors is needed specifically for the F-test's exact null distributionStatistical InferenceMethods of Estimation, Properties of EstimatorsHard
- 148Estimability from a partial set of linear relationsLinear Models & RegressionGauss-Markov Models, Estimability, Best Linear Unbiased EstimatorsMedium
- 149Filling in a two-way ANOVA table from its own consistency constraintsLinear Models & RegressionAnalysis of Variance and Covariance, Fixed/Random/Mixed EffectsMedium
- 150A strongly-supported null hypothesis means a large p-valueStatistical InferenceTests of Hypotheses: MP and UMP Tests, Likelihood Ratio TestsEasy
- 151What the least-squares regression line can and cannot claimLinear Models & RegressionSimple and Multiple Linear Regression, Logistic RegressionMedium
- 152Unbiasedness of the three mean squares in multiple regressionLinear Models & RegressionSimple and Multiple Linear Regression, Logistic RegressionHard
- 153sum yi=sumhat yi holds precisely when there's an interceptLinear Models & RegressionSimple and Multiple Linear Regression, Logistic RegressionMedium
- 154The logistic regression response is binaryLinear Models & RegressionSimple and Multiple Linear Regression, Logistic RegressionEasy
- 155Conditional variance from a block-diagonal covarianceProbabilityRandom Variables and Distribution Functions, Expectation and MomentsEasy
- 156Distribution of a linear combination of correlated bivariate normal componentsProbabilityRandom Variables and Distribution Functions, Expectation and MomentsEasy
- 157The likelihood ratio test for a multivariate mean — flagged against standard Hotelling's T^2/F theoryMultivariate AnalysisMultivariate Normal Distribution, Wishart Distribution, Quadratic FormsHard
- 158Checking positive definiteness of a quadratic form via leading minorsLinear AlgebraQuadratic Forms, Reduction and ClassificationEasy
- 159A mixed-sign quadratic form is never chi-squareStatistical InferenceAnalysis of Discrete Data and Chi-Square Test of Goodness of FitEasy
- 160Computing a partial correlation from three simple correlationsMultivariate AnalysisPartial and Multiple Correlation CoefficientsEasy
- 161A Mahalanobis distance that matches none of the printed optionsMultivariate AnalysisMultivariate Normal Distribution, Wishart Distribution, Quadratic FormsHard
- 162The standard dimension-reduction techniqueMultivariate AnalysisData Reduction: PCA, Discriminant, Cluster AnalysisEasy
- 163Systematic sampling — equal selection probability, without a guaranteed efficiency edgeSampling TheorySimple Random, Stratified and Systematic SamplingHard
- 164Computing inclusion probabilities from an explicit sampling designSampling TheoryProbability Proportional to Size SamplingMedium
- 165The standard inclusion-probability bound for Midzuno's designSampling TheorySimple Random, Stratified and Systematic SamplingEasy
- 166Post-stratification — stratifying after the factSampling TheorySimple Random, Stratified and Systematic SamplingEasy
- 167Double sampling — filling a gap in auxiliary-variable coverageSampling TheoryRatio and Regression MethodsMedium
- 168BLUE of beta2 from three uncorrelated, homoscedastic linear combinationsStatistical InferenceMethods of Estimation, Properties of EstimatorsMedium
- 169BIBD incidence matrix — symmetry only for the special case v=bDesign of ExperimentsConnectedness and Orthogonality of Block Designs, BIBDHard
- 170Confounding a 2^n design into 2^r blocks — how many effects are sacrificedDesign of Experiments2^K Factorial Experiments: Confounding and ConstructionEasy
- 171Error degrees of freedom for an unbalanced CRDDesign of ExperimentsCompletely Randomized, Randomized Block and Latin-Square DesignsEasy
- 172Opposite-signed simple effects average to a zero main effectProbabilityRandom Variables and Distribution Functions, Expectation and MomentsEasy
- 173Probability one exponential competitor fails before anotherSurvival AnalysisHazard Function and Failure Rates, Censoring and Life TestingEasy
- 174Recovering the CDF from a linear hazard functionSurvival AnalysisHazard Function and Failure Rates, Censoring and Life TestingEasy
- 175Surviving past twice the mean lifetime, for an exponential systemSurvival AnalysisHazard Function and Failure Rates, Censoring and Life TestingEasy
- 176Matching a primal and its stated dual — an unresolvable coefficient clashOperations ResearchLinear Programming Problem, Simplex Method, DualityHard
- 177Unique primal optimum implies a unique dual optimum tooOperations ResearchLinear Programming Problem, Simplex Method, DualityHard
- 178Total unimodularity gives the assignment problem its binary solutionsOperations ResearchLinear Programming Problem, Simplex Method, DualityEasy
- 179M/M/1 — spotting the false relation among the standard formulasOperations ResearchSteady-State Solutions of Markovian Queuing Models (M/M/1, M/M/C, M/G/1)Easy
- 180Traffic density for an M/M/3 systemOperations ResearchSteady-State Solutions of Markovian Queuing Models (M/M/1, M/M/C, M/G/1)Easy