STA104/MTH135: Probability

Homework #10: Limits of Random Variables

Due: Tuesday, Dec 2

Text Problems are from Sheldon Ross, A First Course in Probability (6th edn).

• Problems (pp. 427ff)
  

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• Exercises (pp. 429ff)

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• Another Problem
  Let U₁, U₂, ... be independent uniformly-distributed random variables on the unit interval [0,1] and, for each n, let Z_n = max(U₁,...,U_n ) be the largest of the first n variables and let S_n = U₁+...+U_n be their sum. Find the limiting probability distribution of the random variables:
  • X_n = n(1-Z_n)
  • Y_n = (S_n-ⁿ/₂)/n^½
  (i.e., find lim Pr[X_nx] and lim Pr[Y_ny] for every x and y, as n->oo)

  Good Luck!

  
  

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