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128 questions · full solutions

Statistics

Estimators, regression intuition, time-series traps, and the standard-error scaling that decides whether a backtest means anything.

  1. 01Variance of a DieMedium · Private
  2. 02Sixty HeadsMedium · Private
  3. 03Carrying the Normal TableEasy
  4. 04Halving the ErrorEasy
  5. 05What the Slope Really IsMedium · Private
  6. 06Diversification ArithmeticMedium · Private
  7. 07π from DartsEasy
  8. 08MLE for an Arrival RateMedium · Private
  9. 09The Biased MaximumHard · Private
  10. 10n or n − 1?Medium · Private
  11. 11Method of Moments in One LineEasy
  12. 12Consistent, Unbiased, or Both?Hard · Private
  13. 13What a p-Value Is NotMedium · Private
  14. 14Ninety-Five Percent of What?Medium · Private
  15. 15How Many Orders for the A/B Test?Medium · Private
  16. 16One Hundred Junk SignalsMedium · Private
  17. 17t or z?Easy
  18. 18Pair Your ExperimentsMedium · Private
  19. 19Covariance by HandMedium · Private
  20. 20Long Two, Short ThreeEasy
  21. 21Flows Predict Returns. Or Do They?Easy
  22. 22The Mean Above the BarHard · Private
  23. 23Best of Two NormalsHard · Private
  24. 24Two Uniforms Past 1.5Medium · Private
  25. 25The R² Beauty ContestMedium · Private
  26. 26The Ghost RegressorHard · Private
  27. 27Unequal Noise, Unequal DamageMedium · Private
  28. 28Why Backtest t-Stats LieHard · Private
  29. 29Why Betas Get Pulled Toward OneMedium · Private
  30. 30Two Random Walks Walk Into a RegressionHard · Private
  31. 31AR(1) Vital SignsMedium · Private
  32. 32Permanent or Transitory?Hard · Private
  33. 33Quiet Days, Loud DaysMedium · Private
  34. 34Error Bars on a Track RecordHard · Private
  35. 35The Price of RobustnessMedium · Private
  36. 36Updating a Hit RateMedium · Private
  37. 37Sizing the Sample for a Target WidthEasy
  38. 38Power of a Fixed-Size ExperimentMedium · Private
  39. 39Covariance of Functions of a NormalHard · Private
  40. 40Correlation of X and X SquaredMedium · Private
  41. 41The Algo That Wins Everywhere but LosesMedium · Private
  42. 42The k-th Smallest of n UniformsMedium · Private
  43. 43The Sophomore Slump, QuantifiedMedium · Private
  44. 44Information Ratio and the Fundamental LawMedium · Private
  45. 45Correlation Doesn't Care About UnitsEasy
  46. 46Error Bars on a Fill RateEasy
  47. 47Three Sigma Without a DistributionEasy
  48. 48MLE Pinned to the BoundaryMedium · Private
  49. 49Who Makes It Into the Resample?Medium · Private
  50. 50The Smallest p-Value Money Can BuyMedium · Private
  51. 51The Posterior Splits the DifferenceMedium · Private
  52. 52Correlated by ConstructionMedium · Private
  53. 53Counting a Rival's OrdersMedium · Private
  54. 54The Slope That Noise AteMedium · Private
  55. 55What a 52% Hit Rate BuysMedium · Private
  56. 56One Number to KeepMedium · Private
  57. 57Two Sharpe-One Strategies, One BookMedium · Private
  58. 58Fitting the Tail Thickness by MomentsMedium · Private
  59. 59Everyone Can't Hedge EveryoneMedium · Private
  60. 60The Fills You Never SawHard · Private
  61. 61The Floor Under Every EstimatorHard · Private
  62. 62Conditioning Your Way to 13x Less VarianceHard · Private
  63. 63A Whole Interval of MLEsHard · Private
  64. 64Why L1 Loves the MedianHard · Private
  65. 65Peeking Once Costs YouHard · Private
  66. 66What Fraction of Your Discoveries Are Junk?Hard · Private
  67. 67Significant and Probably FalseHard · Private
  68. 68Same Data, Two p-ValuesHard · Private
  69. 69How Long to Certify a 0.05 CorrelationHard · Private
  70. 70Thirty-One Years to Prove ItHard · Private
  71. 71Error Bars for an Option SellerHard · Private
  72. 72Good Managers Lose OftenHard · Private
  73. 73Root-T Breaks When Returns TrendHard · Private
  74. 74The Beta That Grows With the ClockHard · Private
  75. 75The R-Squared You Get for FreeHard · Private
  76. 76Weight the Quiet DaysHard · Private
  77. 77Regress Out the Market FirstHard · Private
  78. 78The Correlation Regime That Wasn'tHard · Private
  79. 79Ticks Sharpen Vol, Never DriftHard · Private
  80. 80Correlations Come in TrianglesHard · Private
  81. 81Two Events That Can't Fully CorrelateHard · Private
  82. 82The Hedge That Shorts the HedgeHard · Private
  83. 83Hedged Within Regimes, Correlated Across ThemHard · Private
  84. 84One Factor to Rule the MatrixHard · Private
  85. 85Fat Tails From Nothing but Vol SwitchingHard · Private
  86. 86Error Bars on a Vol MarkHard · Private
  87. 87How Wobbly Is an Empirical VaR?Hard · Private
  88. 88Six Breaches and a VerdictHard · Private
  89. 89Correlated Until It MattersHard · Private
  90. 90Shrinking Ten Traders at OnceHard · Private
  91. 91Your Best Day Means NothingHard · Private
  92. 92Bootstrapping Away the AutocorrelationHard · Private
  93. 93The Free Lunch in Mirrored DrawsHard · Private
  94. 94Factors From Thin AirHard · Private
  95. 95Five Bad DaysEasy
  96. 96Most Months Are Below AverageMedium · Private
  97. 97Forty-Five Percent, Still Not ProvenMedium · Private
  98. 98Two Extra KnobsMedium · Private
  99. 99Is the Router Playing Favorites?Medium · Private
  100. 100The Next Two RFQsMedium · Private
  101. 101How Fast Does the Edge Rot?Medium · Private
  102. 102One Print to Call the RegimeHard · Private
  103. 103Twin Tripwires at 19 to 1Hard · Private
  104. 104The 1.96 That Wasn'tHard · Private
  105. 105Selected Into AnticorrelationHard · Private
  106. 106Don't Control for the MessengerHard · Private
  107. 107Loud Noise on the Big DaysHard · Private
  108. 108Twenty Days Out, Two Different WorldsHard · Private
  109. 109Whose Correlation Is It?Hard · Private
  110. 110How Hard to Lean on the Peer GroupHard · Private
  111. 111The Optimizer Grades Its Own HomeworkHard · Private
  112. 112Aim Where the Tail LivesHard · Private
  113. 113Subtract What You Already KnowHard · Private
  114. 114The Information in a Day's RangeHard · Private
  115. 115The Second-Worst Day Is Not Your VaRHard · Private
  116. 116When the Prior Stops MatteringHard · Private
  117. 117Diversification Raised My VaRHard · Private
  118. 118The Size That Maximizes the TakeHard · Private
  119. 119Skew Arrives With the RegimeHard · Private
  120. 120Patching the Error Bar, Lag by LagHard · Private
  121. 121Three New Signals, Worse Than NoiseHard · Private
  122. 122Vol That Ignores the Fat FingerHard · Private
  123. 123Proving the Hedge DecayedHard · Private
  124. 124The Correction That Flips the VerdictHard · Private
  125. 125The Overlap IllusionHard · Private
  126. 126Does the Second Bet Help?Medium · Private
  127. 127Add a Third EngineHard · Private
  128. 128The Best of Twenty BacktestsHard · Private