Mathematical Statistics 2016
60 questions, booklet MS2016.
Section A · Section A · Q1–30
Practise this section →- 1Rank of a four by four matrix with dependent rowsMatrices and DeterminantsRank of a matrixMedium
- 2Conditions for a two by two matrix to be its own inverseMatrices and DeterminantsProperties of matricesMedium
- 3Volume of a solid of revolution of a hyperbolaCalculusIntegral calculusMedium
- 4Volume of a region enclosed by a cylinder and planesCalculusIntegral calculusMedium
- 5Probability at most one from a Poisson mgfUnivariate DistributionsPoisson distributionEasy
- 6Probability of an event from conditional probabilities of independent eventsDescriptive Statistics and ProbabilityProbabilityMedium
- 7Existence of a second moment for a power-law tailUnivariate DistributionsMathematical expectation and momentsHard
- 8Chi-square law from a transformed power-law variableUnivariate DistributionsTransformations of random variablesHard
- 9Limit in probability of the mean of squared normalsLimit TheoremsWeak law of large numbersEasy
- 10Conditions for a linear estimator of the slope to be unbiasedEstimationUnbiased estimatorsMedium
- 11Conditional probability given X from a joint densityMultivariate DistributionsConditional distributionsMedium
- 12Asymptotic standard deviation of the mean of oddsLimit TheoremsCentral Limit Theorem (i.i.d. and finite variance case)Hard
- 13Bias and consistency of the sample minimumEstimationConsistency of estimatorsMedium
- 14Limit of the normalised maximum of exponential variablesLimit TheoremsAsymptotic distributionsHard
- 15Probability that a sum of two squared normals lies below fourSampling DistributionsChi-square distributionEasy
- 16Expectation from a piecewise cumulative distribution functionUnivariate DistributionsDistribution functionsMedium
- 17Critical region of the most powerful test for a Laplace scaleTesting of HypothesesMost powerful testsMedium
- 18Distribution of a studentised difference of two observationsSampling Distributionst distributionHard
- 19UMVUE of the square of a Poisson parameterEstimationUMVUEHard
- 20Size and power of a test on a four-point variableTesting of HypothesesType I and Type II errorsMedium
- 21Consistent estimator of theta for a normal with mean twice the sdEstimationConsistency of estimatorsMedium
- 22Posterior probability of a vendor given a long lifetimeDescriptive Statistics and ProbabilityBayes theoremMedium
- 23Solution of a first-order linear differential equationDifferential EquationsFirst order linear equationsMedium
- 24Convergence of two series with exponential decaySequences and SeriesConvergence of seriesMedium
- 25Differentiability at zero of two oscillating functionsCalculusDifferential calculus of one variableHard
- 26Mean value theorem on adjacent intervalsCalculusDifferential calculus of one variableHard
- 27Local minima of a cubic polynomial in two variablesCalculusDifferential calculus of two variablesHard
- 28Solution of a second-order constant coefficient equationDifferential EquationsGeneral theory of linear ODEsMedium
- 29Area under the second derivative of a convolution integralCalculusIntegral calculusMedium
- 30Matrix identity from known eigenvaluesMatrices and DeterminantsCayley-Hamilton theoremHard
Section B · Section B · Q31–40
Practise this section →- 31Solutions of a symmetric two by two linear systemMatrices and DeterminantsSystems of linear equationsMedium
- 32Sequence and series convergence for a two-branch sequenceSequences and SeriesConvergence of seriesMedium
- 33Limits of powers times a function with a cancelling expansionCalculusDifferential calculus of one variableHard
- 34Continuity at a point for functions involving a floor termCalculusDifferential calculus of one variableHard
- 35Conditional probability inequalities from positive associationDescriptive Statistics and ProbabilityProbabilityMedium
- 36Non-unique MLE for a uniform distribution of fixed widthEstimationMaximum likelihood estimationHard
- 37Testing problems with a large-sum rejection regionTesting of HypothesesMost powerful testsMedium
- 38Sufficiency, completeness and UMVUE for a normal variance parameterEstimationSufficiency and completenessHard
- 39Confidence intervals for an exponential rate parameterEstimationConfidence intervalsMedium
- 40Properties of a distribution function with a jump and a continuous partUnivariate DistributionsDistribution functionsMedium
Section C · Section C · Q41–60
Practise this section →- 41Variance scaling for a difference of two sample meansSampling DistributionsSampling distributions from normalMedium
- 42Chebyshev upper bound for a triangular densityUnivariate DistributionsChebyshev inequalityMedium
- 43Probability that X is less than Y for i.i.d. variablesMultivariate DistributionsBivariate distributionsEasy
- 44Quadrant probability for independent standard normalsMultivariate DistributionsBivariate normal distributionEasy
- 45Normal approximation to a binomial probabilityLimit TheoremsCentral Limit Theorem (i.i.d. and finite variance case)Medium
- 46Probability of at least one success from a binomial with shared parameterUnivariate DistributionsBinomial distributionMedium
- 47Rank of a linear transformation from its matrixMatrices and DeterminantsRank of a matrixMedium
- 48Limit of n times a bounded exponentially damped term plus a sineSequences and SeriesConvergence of sequencesMedium
- 49Minimum of an increasing function on an intervalCalculusDifferential calculus of one variableMedium
- 50Arc length from a given derivativeCalculusIntegral calculusMedium
- 51Value of the upper limit from a triple integralCalculusIntegral calculusHard
- 52Product of the eigenvalues of the inverse of a matrixMatrices and DeterminantsDeterminantsMedium
- 53Lower angle limit from a polar double integralCalculusIntegral calculusMedium
- 54Telescoping series in reciprocals of squaresSequences and SeriesInfinite series and its convergenceMedium
- 55Normal approximation for a sum of gamma variablesLimit TheoremsCentral Limit Theorem (i.i.d. and finite variance case)Medium
- 56Mean of a variable from a factorising joint densityMultivariate DistributionsJoint densitiesMedium
- 57Probability that two random points are close togetherUnivariate DistributionsUniform distributionMedium
- 58Probability that the square of a variable lies in an intervalUnivariate DistributionsContinuous distributionsMedium
- 59Probability that a uniform point is near an end of the intervalUnivariate DistributionsUniform distributionEasy
- 60Probability that a family has children of both sexesDescriptive Statistics and ProbabilityProbabilityHard