The lim-inf equalsthe limCan't useDCTbecause thesum may notbe L¹Prove itfor simpleRVs firstWrite ℤ-valued RVas a sum ofindicatorfunctionsUseenumeration ofℚ or stateseparability as anecessaryconditionIdentically distributedRVs have the sameexpectation/varianceA result for allintervals [-∞, b]impliessomethingabout allintervals [a,b]Probabilitythat arandom walkwill exit [a,b]at b is b/(b-a)Prove integralsare equal byintegrating theirdifferenceWald's2ⁿᵈequationSomethinglegitimatelyrelevant toVegasArithmeticwith 0 and ∞to derive acontradictionA is the smallest___ containing B;and C is a ___containing B;therefore Ccontains A.Flippanttreatment ofa measure-0setStandardize aRV, then use acalculation forthe standardversionπ-λtheoremDo somethingwith anonnegative RVthat would notwork for a real-valued RVPull sum ofindicatorfunctions out of(or into) anintegral viaBeppo LevyWald's1ˢᵗequationThe nᵗʰincrement Xₙof a RW isindependentfrom Sₙ₋₁L²⊆L¹Showintegral is0 usingFatouFor integersx, y: thenegation ofx≥y is x≤(y-1)Convergencein probabilityThe lim-inf equalsthe limCan't useDCTbecause thesum may notbe L¹Prove itfor simpleRVs firstWrite ℤ-valued RVas a sum ofindicatorfunctionsUseenumeration ofℚ or stateseparability as anecessaryconditionIdentically distributedRVs have the sameexpectation/varianceA result for allintervals [-∞, b]impliessomethingabout allintervals [a,b]Probabilitythat arandom walkwill exit [a,b]at b is b/(b-a)Prove integralsare equal byintegrating theirdifferenceWald's2ⁿᵈequationSomethinglegitimatelyrelevant toVegasArithmeticwith 0 and ∞to derive acontradictionA is the smallest___ containing B;and C is a ___containing B;therefore Ccontains A.Flippanttreatment ofa measure-0setStandardize aRV, then use acalculation forthe standardversionπ-λtheoremDo somethingwith anonnegative RVthat would notwork for a real-valued RVPull sum ofindicatorfunctions out of(or into) anintegral viaBeppo LevyWald's1ˢᵗequationThe nᵗʰincrement Xₙof a RW isindependentfrom Sₙ₋₁L²⊆L¹Showintegral is0 usingFatouFor integersx, y: thenegation ofx≥y is x≤(y-1)Convergencein probability

Math 5530H Bingo 1 - Call List

(Print) Use this randomly generated list as your call list when playing the game. There is no need to say the BINGO column name. Place some kind of mark (like an X, a checkmark, a dot, tally mark, etc) on each cell as you announce it, to keep track. You can also cut out each item, place them in a bag and pull words from the bag.


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  1. The lim-inf equals the lim
  2. Can't use DCT because the sum may not be L¹
  3. Prove it for simple RVs first
  4. Write ℤ-valued RV as a sum of indicator functions
  5. Use enumeration of ℚ or state separability as a necessary condition
  6. Identically distributed RVs have the same expectation/variance
  7. A result for all intervals [-∞, b] implies something about all intervals [a,b]
  8. Probability that a random walk will exit [a,b] at b is b/(b-a)
  9. Prove integrals are equal by integrating their difference
  10. Wald's 2ⁿᵈ equation
  11. Something legitimately relevant to Vegas
  12. Arithmetic with 0 and ∞ to derive a contradiction
  13. A is the smallest ___ containing B; and C is a ___ containing B; therefore C contains A.
  14. Flippant treatment of a measure-0 set
  15. Standardize a RV, then use a calculation for the standard version
  16. π-λ theorem
  17. Do something with a nonnegative RV that would not work for a real-valued RV
  18. Pull sum of indicator functions out of (or into) an integral via Beppo Levy
  19. Wald's 1ˢᵗ equation
  20. The nᵗʰ increment Xₙ of a RW is independent from Sₙ₋₁
  21. L²⊆L¹
  22. Show integral is 0 using Fatou
  23. For integers x, y: the negation of x≥y is x≤(y-1)
  24. Convergence in probability