Probability
192 questions across 4 topics. Also examined in CNET, ISS, GATE ST, GATE DA, JAM MS.
Random Variables and Distribution Functions, Expectation and Moments
- 105Conditional distribution of a binomial given the sumMSET June 2025Random Variables and Distribution Functions, Expectation and MomentsHard
- 110Which transformations of a c.d.f. are c.d.f.sMSET June 2025Random Variables and Distribution Functions, Expectation and MomentsHard
- 111Symmetry about the medianMSET June 2025Random Variables and Distribution Functions, Expectation and MomentsHard
- 116Indicators of vanishing eventsMSET June 2025Random Variables and Distribution Functions, Expectation and MomentsHard
- 117Slutsky's theoremMSET June 2025Random Variables and Distribution Functions, Expectation and MomentsHard
- 124The Yule–Furry pure birth processMSET June 2025Random Variables and Distribution Functions, Expectation and MomentsHard
- 125Normalising a geometric-type p.m.f. on even integersMSET June 2025Random Variables and Distribution Functions, Expectation and MomentsMedium
- 126Probability generating function of a linear transformMSET June 2025Random Variables and Distribution Functions, Expectation and MomentsMedium
- 128An m.g.f. identity forcing symmetry about 5MSET June 2025Random Variables and Distribution Functions, Expectation and MomentsHard
- 129Expectation from a c.d.f. — discrete and continuousMSET June 2025Random Variables and Distribution Functions, Expectation and MomentsHard
- 130Ratios of exponentials — F and BetaMSET June 2025Random Variables and Distribution Functions, Expectation and MomentsHard
- 131Sufficiency for the Bernoulli parameterMSET June 2025Random Variables and Distribution Functions, Expectation and MomentsHard
- 148Identifying estimable functions from the designMSET June 2025Random Variables and Distribution Functions, Expectation and MomentsHard
- 155A conditional variance that is really an independence checkMSET June 2025Random Variables and Distribution Functions, Expectation and MomentsHard
- 156Conditional distribution in a trivariate normalMSET June 2025Random Variables and Distribution Functions, Expectation and MomentsHard
- 160Feasible range of a third correlationMSET June 2025Random Variables and Distribution Functions, Expectation and MomentsHard
- 167When a stratum deserves a larger sampleMSET June 2025Random Variables and Distribution Functions, Expectation and MomentsHard
- 106Measurability — what forward images do not preserveMSET April 2024Random Variables and Distribution Functions, Expectation and MomentsHard
- 107Measurability — functions relative to a coarse sigma-fieldMSET April 2024Random Variables and Distribution Functions, Expectation and MomentsHard
- 108Classical probability — pairwise independence and a unionMSET April 2024Random Variables and Distribution Functions, Expectation and MomentsHard
- 109Joint distributions — a table that happens to factorMSET April 2024Random Variables and Distribution Functions, Expectation and MomentsHard
- 110Independent testers — a false "all-of-the-above"MSET April 2024Random Variables and Distribution Functions, Expectation and MomentsHard
- 112Markov's inequality — applied to the fourth momentMSET April 2024Random Variables and Distribution Functions, Expectation and MomentsEasy
- 116Scaling a fixed random variable — convergence is to X, not to 0MSET April 2024Random Variables and Distribution Functions, Expectation and MomentsHard
- 125Matching means — Binomial is always less dispersed than PoissonMSET April 2024Random Variables and Distribution Functions, Expectation and MomentsEasy
- 129A shift identity — the mean of a truncated variableMSET April 2024Random Variables and Distribution Functions, Expectation and MomentsHard
- 130Geometric distribution — solving for p from the varianceMSET April 2024Random Variables and Distribution Functions, Expectation and MomentsEasy
- 137Setting up a one-sided test — the claim goes in H1MSET April 2024Random Variables and Distribution Functions, Expectation and MomentsMedium
- 138Exponential families — spotting the curved caseMSET April 2024Random Variables and Distribution Functions, Expectation and MomentsHard
- 148Estimable functions — a rank-deficient three-parameter modelMSET April 2024Random Variables and Distribution Functions, Expectation and MomentsHard
- 156Bivariate normal — shape of the conditional mean and varianceMSET April 2024Random Variables and Distribution Functions, Expectation and MomentsMedium
- 158Quadratic forms — checking positive definiteness via leading minorsMSET April 2024Random Variables and Distribution Functions, Expectation and MomentsMedium
- 159A quadratic form that collapses to a perfect squareMSET April 2024Random Variables and Distribution Functions, Expectation and MomentsMedium
- 164Inclusion probabilities — they always sum to the expected sample sizeMSET April 2024Random Variables and Distribution Functions, Expectation and MomentsHard
- 101Recognising a binomial PGFMSET March 2023Random Variables and Distribution Functions, Expectation and MomentsEasy
- 105Four probability facts — only subadditivity survives unconditionallyMSET March 2023Random Variables and Distribution Functions, Expectation and MomentsHard
- 106Reading probability off a piecewise CDF, including a jumpMSET March 2023Random Variables and Distribution Functions, Expectation and MomentsMedium
- 107A false moment inequality — checked by direct counterexampleMSET March 2023Random Variables and Distribution Functions, Expectation and MomentsMedium
- 108Two indicator variables that are neither identically distributed nor independentMSET March 2023Random Variables and Distribution Functions, Expectation and MomentsMedium
- 109Reading facts off an empirical distribution from raw dataMSET March 2023Random Variables and Distribution Functions, Expectation and MomentsMedium
- 110A four-point discrete distribution built from degenerate mixturesMSET March 2023Random Variables and Distribution Functions, Expectation and MomentsMedium
- 111Law of total variance for a mixed Binomial–Geometric hierarchyMSET March 2023Random Variables and Distribution Functions, Expectation and MomentsMedium
- 116Multiplying a fixed random variable by an indicator converging to 1MSET March 2023Random Variables and Distribution Functions, Expectation and MomentsMedium
- 117Slutsky's theorem — convergence in distribution is all that's guaranteedMSET March 2023Random Variables and Distribution Functions, Expectation and MomentsHard
- 125A Poisson approximation for rare disease counts in a large new sampleMSET March 2023Random Variables and Distribution Functions, Expectation and MomentsMedium
- 126Expected maximum of two iid standard normalsMSET March 2023Random Variables and Distribution Functions, Expectation and MomentsEasy
- 127Finding the PGF that gives variance exactly 1MSET March 2023Random Variables and Distribution Functions, Expectation and MomentsMedium
- 129A symmetry argument for the conditional mean of an exponential splitMSET March 2023Random Variables and Distribution Functions, Expectation and MomentsMedium
- 135Minimal sufficiency survives under a bijective relabellingMSET March 2023Random Variables and Distribution Functions, Expectation and MomentsHard
- 140Asymptotic variance of the sample median for normal dataMSET March 2023Random Variables and Distribution Functions, Expectation and MomentsMedium
- 142The combinatorial definition of the Mann–Whitney U statisticMSET March 2023Random Variables and Distribution Functions, Expectation and MomentsHard
- 143Two standard Fisher information constantsMSET March 2023Random Variables and Distribution Functions, Expectation and MomentsMedium
- 144Matched-pair design calls for the paired t-testMSET March 2023Random Variables and Distribution Functions, Expectation and MomentsMedium
- 155Conditional variance from a block-diagonal covarianceMSET March 2023Random Variables and Distribution Functions, Expectation and MomentsEasy
- 156Distribution of a linear combination of correlated bivariate normal componentsMSET March 2023Random Variables and Distribution Functions, Expectation and MomentsEasy
- 172Opposite-signed simple effects average to a zero main effectMSET March 2023Random Variables and Distribution Functions, Expectation and MomentsEasy
- 102Coefficient of variation — invariant under rescalingMSET September 2021Random Variables and Distribution Functions, Expectation and MomentsEasy
- 105Markov's inequality on the third absolute momentMSET September 2021Random Variables and Distribution Functions, Expectation and MomentsEasy
- 106Reading a probability off a joint tableMSET September 2021Random Variables and Distribution Functions, Expectation and MomentsMedium
- 118The classical assumptions behind the two-sample t-testMSET September 2021Random Variables and Distribution Functions, Expectation and MomentsMedium
- 124A newsvendor problem with an unstated order quantityMSET September 2021Random Variables and Distribution Functions, Expectation and MomentsHard
- 132Measurability between two explicit finite sigma-fieldsMSET September 2021Random Variables and Distribution Functions, Expectation and MomentsHard
- 133A Gaussian-weighted measure — absolutely continuous both waysMSET September 2021Random Variables and Distribution Functions, Expectation and MomentsHard
- 134Complete convergence implies almost-sure convergence, with no independence neededMSET September 2021Random Variables and Distribution Functions, Expectation and MomentsHard
- 136What convergence in distribution does and does not carry overMSET September 2021Random Variables and Distribution Functions, Expectation and MomentsHard
- 138Kolmogorov's 0–1 law — convergence of an independent series is a tail eventMSET September 2021Random Variables and Distribution Functions, Expectation and MomentsHard
- 140Decomposing a mixed distribution function into its discrete and continuous partsMSET September 2021Random Variables and Distribution Functions, Expectation and MomentsHard
- 141Discontinuities of a CDF are always at most countableMSET September 2021Random Variables and Distribution Functions, Expectation and MomentsEasy
- 154Asymptotic relative efficiency of the median versus a scaled mean for Uniform(0,theta)MSET September 2021Random Variables and Distribution Functions, Expectation and MomentsMedium
- 165Orthogonal design columns don't automatically make sums of squares uncorrelatedMSET September 2021Random Variables and Distribution Functions, Expectation and MomentsHard
- 173Properties of an autocovariance function — spotting the reversed inequalityMSET September 2021Random Variables and Distribution Functions, Expectation and MomentsHard
- 175AR(1) with an intercept — the mean is not simply muMSET September 2021Random Variables and Distribution Functions, Expectation and MomentsMedium
- 7Normalizing a quadratic density and integrating past x=1MSET December 2020Random Variables and Distribution Functions, Expectation and MomentsEasy
- 105Disjoint events are never (nondegenerately) independentMSET December 2020Random Variables and Distribution Functions, Expectation and MomentsMedium
- 107Reading a joint pmf table directly to find P[X=2Y]MSET December 2020Random Variables and Distribution Functions, Expectation and MomentsHard
- 109The reciprocal of a Cauchy is Cauchy, with a location/scale inversion formulaMSET December 2020Random Variables and Distribution Functions, Expectation and MomentsMedium
- 110Poisson probability of at least one error in three times the base intervalMSET December 2020Random Variables and Distribution Functions, Expectation and MomentsEasy
- 132A crude finite/infinite indicator set function fails finite additivityMSET December 2020Random Variables and Distribution Functions, Expectation and MomentsHard
- 133The exponential measure is absolutely continuous w.r.t. Lebesgue measure, never the reverseMSET December 2020Random Variables and Distribution Functions, Expectation and MomentsHard
- 135Two limits that quote the wrong constant: E[X^2] and Var(X), not 2 and 0MSET December 2020Random Variables and Distribution Functions, Expectation and MomentsHard
- 136Almost-sure convergence transfers through bounded continuous maps, but not through the unbounded exponentialMSET December 2020Random Variables and Distribution Functions, Expectation and MomentsHard
- 138Kolmogorov's convergence theorem: summable variances force a.s. convergenceMSET December 2020Random Variables and Distribution Functions, Expectation and MomentsEasy
- 140Computing E(X) for a mixed distribution: two atoms plus a continuous pieceMSET December 2020Random Variables and Distribution Functions, Expectation and MomentsMedium
- 141A monotone function has at most countably many jumps, leaving uncountably many continuity pointsMSET December 2020Random Variables and Distribution Functions, Expectation and MomentsEasy
- 142The Minkowski inequality for the L^3 normMSET December 2020Random Variables and Distribution Functions, Expectation and MomentsEasy
- 144The law of total covariance shows X and Y share only a fraction of their variability — never perfect correlationMSET December 2020Random Variables and Distribution Functions, Expectation and MomentsMedium
- 149The score function has mean zero under standard regularityMSET December 2020Random Variables and Distribution Functions, Expectation and MomentsEasy
- 157Eigenvector rows diagonalize Sigma, guaranteeing independent marginalsMSET December 2020Random Variables and Distribution Functions, Expectation and MomentsMedium
- 159The standard multivariate-normal conditional mean and covariance formulasMSET December 2020Random Variables and Distribution Functions, Expectation and MomentsHard
- 161Solving the normal equations for an exactly-identified three-parameter, three-observation modelMSET December 2020Random Variables and Distribution Functions, Expectation and MomentsHard
- 174Variance of the mean of ten equicorrelated unit-variance variablesMSET December 2020Random Variables and Distribution Functions, Expectation and MomentsHard
- 175Only one candidate MSE formula wrongly tacks on an extra mu^2 termMSET December 2020Random Variables and Distribution Functions, Expectation and MomentsHard
- 102Covariance is bilinear: shifts vanish, scalars multiply outMSET June 2019Random Variables and Distribution Functions, Expectation and MomentsEasy
- 103A common variance means quantile ordering is decided purely by the meansMSET June 2019Random Variables and Distribution Functions, Expectation and MomentsMedium
- 105Bounding E[X] by the range of the free probability massMSET June 2019Random Variables and Distribution Functions, Expectation and MomentsEasy
- 106Independence lets a compound conditioning event drop outMSET June 2019Random Variables and Distribution Functions, Expectation and MomentsEasy
- 107MGF of a sum multiplies; MGF of a product needs conditioningMSET June 2019Random Variables and Distribution Functions, Expectation and MomentsMedium
- 108E[YX]=X^2 for standard normal X forces zero correlation, not positiveMSET June 2019Random Variables and Distribution Functions, Expectation and MomentsEasy
- 111A mixed discrete-continuous joint law, checked term by termMSET June 2019Random Variables and Distribution Functions, Expectation and MomentsHard
- 131Finite additivity alone gives superadditivity over countable disjoint unionsMSET June 2019Random Variables and Distribution Functions, Expectation and MomentsMedium
- 132A bounded sequence is not dominated by an integrable function on an infinite measure spaceMSET June 2019Random Variables and Distribution Functions, Expectation and MomentsMedium
- 133A measurable map must be constant on every atom of the sigma-fieldMSET June 2019Random Variables and Distribution Functions, Expectation and MomentsMedium
- 136Here Y is uniform, so Xmid Y=1 is a unit exponentialMSET June 2019Random Variables and Distribution Functions, Expectation and MomentsMedium
- 140A distribution function with two jumps and a uniform stretchMSET June 2019Random Variables and Distribution Functions, Expectation and MomentsMedium
- 141A mixed law: uniform density tfrac12 on [0,1) plus an atom of tfrac12 at 1MSET June 2019Random Variables and Distribution Functions, Expectation and MomentsEasy
- 173The random-phase sinusoid has autocovariance tfrac12cos(omega h)MSET June 2019Random Variables and Distribution Functions, Expectation and MomentsMedium
- 174A random walk's variance grows linearly, so only Xt/sqrt t has stable varianceMSET June 2019Random Variables and Distribution Functions, Expectation and MomentsMedium
- 175Variance of a mean of three AR(1) values needs autocovariances, not autocorrelationsMSET June 2019Random Variables and Distribution Functions, Expectation and MomentsHard
- 9The distribution function has atoms of tfrac14 at 2 and 3 — no option matchesMSET January 2018 Paper IIRandom Variables and Distribution Functions, Expectation and MomentsMedium
- 54X+Y is 2X on Xlealpha and exactly 0 otherwise: an atom at the originMSET January 2018 Paper IIRandom Variables and Distribution Functions, Expectation and MomentsMedium
- 56Change of variables for Y=1/X: the Jacobian is 1/y^2MSET January 2018 Paper IIRandom Variables and Distribution Functions, Expectation and MomentsEasy
- 57Exchangeability gives P(X1le X2)=tfrac12 for a continuous FMSET January 2018 Paper IIRandom Variables and Distribution Functions, Expectation and MomentsEasy
- 58Y=X^2 has density y^-1/2/(2theta) on (0,theta^2) — option (C) misprints the rangeMSET January 2018 Paper IIRandom Variables and Distribution Functions, Expectation and MomentsEasy
- 63Cov(bar X,X1-X2)=(Var X1-Var X2)/n=0MSET January 2018 Paper IIRandom Variables and Distribution Functions, Expectation and MomentsEasy
- 80A normal scale mixture with a chi^21 precision: X is CauchyMSET January 2018 Paper IIIRandom Variables and Distribution Functions, Expectation and MomentsHard
- 85E(Umid F) and E(Vmid F) are both X — hardly independentMSET January 2018 Paper IIIRandom Variables and Distribution Functions, Expectation and MomentsEasy
- 143Density tfrac12 on (0,1) plus atoms of tfrac14 at 1 and 2MSET January 2018 Paper IIIRandom Variables and Distribution Functions, Expectation and MomentsEasy
- 144Uniform on the triangle 0<x<y<1: given Y=y, Xsim U(0,y)MSET January 2018 Paper IIIRandom Variables and Distribution Functions, Expectation and MomentsEasy
- 10P(X2<X1)=theta1/(2theta2) when theta1<theta2MSET April 2017 Paper IIRandom Variables and Distribution Functions, Expectation and MomentsMedium
- 56Y=X^2 collects both roots, with Jacobian 1/(2sqrt y)MSET April 2017 Paper IIRandom Variables and Distribution Functions, Expectation and MomentsEasy
- 84Flipping the sign outside [-alpha,alpha] preserves the N(0,1) lawMSET April 2017 Paper IIIRandom Variables and Distribution Functions, Expectation and MomentsEasy
- 86Atoms tfrac14 at -1 and tfrac12 at 2, plus density tfrac14 on (2,3)MSET April 2017 Paper IIIRandom Variables and Distribution Functions, Expectation and MomentsMedium
- 87Atoms of tfrac13 at 0 and 1 with density tfrac13 betweenMSET April 2017 Paper IIIRandom Variables and Distribution Functions, Expectation and MomentsMedium
- 88Mixture mean: 0.4times5+0.6times7MSET April 2017 Paper IIIRandom Variables and Distribution Functions, Expectation and MomentsEasy
- 901-X is Bernoulli(0.4), so F(0.65)=P(1-X=0)=0.6MSET April 2017 Paper IIIRandom Variables and Distribution Functions, Expectation and MomentsEasy
- 91Density x+y on the unit square: Cov(X,Y)=-tfrac1144MSET April 2017 Paper IIIRandom Variables and Distribution Functions, Expectation and MomentsMedium
- 129Started at 0, the AR(1) has mean mu1-phi^t/1-phi and variance frac1-phi^2t1-phi^2MSET April 2017 Paper IIIRandom Variables and Distribution Functions, Expectation and MomentsMedium
- 139The residual Y-E(Ymid X) is uncorrelated with every function of XMSET April 2017 Paper IIIRandom Variables and Distribution Functions, Expectation and MomentsEasy
- 145EX-Y for uniform marginals never exceeds tfrac12MSET April 2017 Paper IIIRandom Variables and Distribution Functions, Expectation and MomentsMedium
Characteristic Functions, Probability Inequalities
- 112Jensen's inequality — direction of the inequalityMSET June 2025Characteristic Functions, Probability InequalitiesHard
- 113Recognising a characteristic functionMSET June 2025Characteristic Functions, Probability InequalitiesHard
- 118What convergence in distribution does and does not giveMSET June 2025Characteristic Functions, Probability InequalitiesHard
- 113Characteristic functions — recognising the standard Laplace lawMSET April 2024Characteristic Functions, Probability InequalitiesMedium
- 112Jensen's inequality applied to four power transformations — x^1/4 is concave, not convexMSET March 2023Characteristic Functions, Probability InequalitiesEasy
- 113Recognising e^-t as the standard Cauchy (= t1) characteristic functionMSET March 2023Characteristic Functions, Probability InequalitiesEasy
- 107Spotting a genuine characteristic function among four candidatesMSET September 2021Characteristic Functions, Probability InequalitiesHard
- 135Rates of convergence for sums of iid mean-zero variables — checking each exponent by ChebyshevMSET September 2021Characteristic Functions, Probability InequalitiesHard
- 137Quadratic-mean convergence — what survives under addition, multiplication and e^i(cdot)MSET September 2021Characteristic Functions, Probability InequalitiesHard
- 142Jensen's inequality for the (concave) logarithmMSET September 2021Characteristic Functions, Probability InequalitiesHard
- 143Lévy's inversion formulaMSET September 2021Characteristic Functions, Probability InequalitiesMedium
- 104E[X^2]<(E[X])^2 would require negative varianceMSET December 2020Characteristic Functions, Probability InequalitiesEasy
- 106E[XYmid X]=X, not the constant 1, for independent unit-mean X,YMSET December 2020Characteristic Functions, Probability InequalitiesMedium
- 143Fourier inversion recovers the density from an integrable characteristic functionMSET December 2020Characteristic Functions, Probability InequalitiesEasy
- 142Markov's inequality applied to g(X)MSET June 2019Characteristic Functions, Probability InequalitiesEasy
- 143The Cauchy characteristic function: e^itmu-sigmatMSET June 2019Characteristic Functions, Probability InequalitiesEasy
- 11The odd term integrates out, leaving X1sim U(-1,1)MSET January 2018 Paper IICharacteristic Functions, Probability InequalitiesMedium
- 12Chebyshev: P[X-EX1]leVar X/1^2MSET January 2018 Paper IICharacteristic Functions, Probability InequalitiesEasy
- 138PGF tfrac12(s+s^2): X is 1 or 2 with equal chanceMSET January 2018 Paper IIICharacteristic Functions, Probability InequalitiesEasy
- 145A point mass at c has characteristic function e^itcMSET January 2018 Paper IIICharacteristic Functions, Probability InequalitiesEasy
- 55Jensen with the strictly concave logarithmMSET April 2017 Paper IICharacteristic Functions, Probability InequalitiesEasy
- 57Independent summands of a constant must both be constantMSET April 2017 Paper IICharacteristic Functions, Probability InequalitiesMedium
- 80-X has characteristic function phi(-t)=overlinephi(t), not -phi(t)MSET April 2017 Paper IIICharacteristic Functions, Probability InequalitiesEasy
- 83Integer values give period 2pi, not piMSET April 2017 Paper IIICharacteristic Functions, Probability InequalitiesEasy
- 89phi(t)=1/(1-it) for the unit exponentialMSET April 2017 Paper IIICharacteristic Functions, Probability InequalitiesEasy
Modes of Convergence, Laws of Large Numbers, Central Limit Theorem
- 114Borel–Cantelli lemmasMSET June 2025Modes of Convergence, Laws of Large Numbers, Central Limit TheoremHard
- 115Continuous mapping theoremMSET June 2025Modes of Convergence, Laws of Large Numbers, Central Limit TheoremHard
- 114Almost-sure vs. pointwise convergence — a single exceptional pointMSET April 2024Modes of Convergence, Laws of Large Numbers, Central Limit TheoremHard
- 115A vanishing-probability spike — Borel–Cantelli settles a.s. convergenceMSET April 2024Modes of Convergence, Laws of Large Numbers, Central Limit TheoremHard
- 117Continuous mapping theorem — probability + distribution still falls short of a.s.MSET April 2024Modes of Convergence, Laws of Large Numbers, Central Limit TheoremHard
- 118Strong law — applied to the squared observationsMSET April 2024Modes of Convergence, Laws of Large Numbers, Central Limit TheoremHard
- 114Borel–Cantelli I and its contrapositive, without needing independenceMSET March 2023Modes of Convergence, Laws of Large Numbers, Central Limit TheoremHard
- 118Rates of growth for sums and sums-of-squares of iid mean-zero unit-variance variablesMSET March 2023Modes of Convergence, Laws of Large Numbers, Central Limit TheoremHard
- 141The sample mean of Cauchy data never concentratesMSET March 2023Modes of Convergence, Laws of Large Numbers, Central Limit TheoremHard
- 155Asymptotic variance of the empirical distribution function at a pointMSET September 2021Modes of Convergence, Laws of Large Numbers, Central Limit TheoremEasy
- 134Products and exponentials of an a.s.-vanishing sequence stay degenerate at their limitsMSET December 2020Modes of Convergence, Laws of Large Numbers, Central Limit TheoremHard
- 137Uniform boundedness with diverging total variance forces the CLT via Lyapunov's conditionMSET December 2020Modes of Convergence, Laws of Large Numbers, Central Limit TheoremHard
- 134A bounded function of i.i.d. variables always obeys the SLLNMSET June 2019Modes of Convergence, Laws of Large Numbers, Central Limit TheoremMedium
- 135CLT for the squares: a finite fourth moment gives a normal limitMSET June 2019Modes of Convergence, Laws of Large Numbers, Central Limit TheoremEasy
- 137Almost-sure convergence does not imply mean-square convergenceMSET June 2019Modes of Convergence, Laws of Large Numbers, Central Limit TheoremEasy
- 55Kolmogorov's 0–1 law for the tail sigma-fieldMSET January 2018 Paper IIModes of Convergence, Laws of Large Numbers, Central Limit TheoremEasy
- 81(1-Y/n^alpha)^n tends to 1-Y, e^-Y or 1 according as alpha<1, =1, 1MSET January 2018 Paper IIIModes of Convergence, Laws of Large Numbers, Central Limit TheoremMedium
- 82Rare large jumps break Lindeberg: the normalised sum is not GaussianMSET January 2018 Paper IIIModes of Convergence, Laws of Large Numbers, Central Limit TheoremHard
- 53Marginal convergence in distribution says nothing about sumsMSET April 2017 Paper IIModes of Convergence, Laws of Large Numbers, Central Limit TheoremEasy
- 54frac1n(n+1)sum jXj tends to E(X1)/2 — the option lacks a factor 2MSET April 2017 Paper IIModes of Convergence, Laws of Large Numbers, Central Limit TheoremMedium
- 85The average winnings settle near m<m/2MSET April 2017 Paper IIIModes of Convergence, Laws of Large Numbers, Central Limit TheoremEasy
Sample Space, Discrete Probability, Independent Events, Bayes Theorem
- 104Bayes' theoremMSET June 2025Sample Space, Discrete Probability, Independent Events, Bayes TheoremMedium
- 107Independence of eventsMSET June 2025Sample Space, Discrete Probability, Independent Events, Bayes TheoremHard
- 108Bayes' theorem — the truthful witnessMSET June 2025Sample Space, Discrete Probability, Independent Events, Bayes TheoremMedium
- 109Total probability and Bayes' theoremMSET June 2025Sample Space, Discrete Probability, Independent Events, Bayes TheoremHard
- 145The Bayes factorMSET June 2025Sample Space, Discrete Probability, Independent Events, Bayes TheoremHard
- 111Events with P(E)<P(F) and overlap — no implication survivesMSET April 2024Sample Space, Discrete Probability, Independent Events, Bayes TheoremHard
- 7A direct conditional-probability computationMSET September 2021Sample Space, Discrete Probability, Independent Events, Bayes TheoremEasy
- 104Bounding P(A) from a conditional probabilityMSET September 2021Sample Space, Discrete Probability, Independent Events, Bayes TheoremEasy
- 108Unpacking a product-form independence hypothesisMSET September 2021Sample Space, Discrete Probability, Independent Events, Bayes TheoremHard
- 8Mutual exclusivity makes "A but not B" just P(A)MSET June 2019Sample Space, Discrete Probability, Independent Events, Bayes TheoremEasy
- 104P(Amid B)=P(B) forces P(B)^2=P(Acap B)le P(A)MSET June 2019Sample Space, Discrete Probability, Independent Events, Bayes TheoremEasy
- 138As printed the double intersection is not a tail event; read as liminf An it isMSET June 2019Sample Space, Discrete Probability, Independent Events, Bayes TheoremHard
- 53A countable intersection of sure events is sureMSET January 2018 Paper IISample Space, Discrete Probability, Independent Events, Bayes TheoremEasy
- 86Given X1+X2=0, exactly one of the two is +1MSET January 2018 Paper IIISample Space, Discrete Probability, Independent Events, Bayes TheoremEasy
- 11P(Acup B)+P(Acup B^c)=1+P(A)MSET April 2017 Paper IISample Space, Discrete Probability, Independent Events, Bayes TheoremEasy
- 12The remaining probability 0.3 splits 2:1MSET April 2017 Paper IISample Space, Discrete Probability, Independent Events, Bayes TheoremEasy
- 13A missing inequality: the stem gives only one of the two comparisons it needsMSET April 2017 Paper IISample Space, Discrete Probability, Independent Events, Bayes TheoremEasy