STA104/MTH135: Probability

Homework #10: Limits of Random Variables

Due: Thursday, Dec 7

Text Problems are from Sheldon Ross, A First Course in Probability (5th edn).

• Problems (pp. 421ff)
  

  4

  5

  6

  10

  11

  13

  15

  18

• Exercises (pp. 424 ff)

  7

• 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!

  
  

  ---