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Unformatted text preview: Final Exam (2 pages) Probability Theory (MATH 235A, Fall 2007) 1. (12 pts) One tosses a fair coin (H=Heads, T=Tails) until the first ap pearance of the pattern HT. How many tosses on average is made? 2. (13 pts) Prove that, for a random variable X with finite variance, one has Var( X ) = min a R E ( X a ) 2 . 3. Consider independent random variables X 1 , X 2 , . . . uniformly distributed in [0 , 1], and let Z n = max k n X k . (a) (8 pts) Prove that Z n 1 in probability. (b) (8 pts) Prove that Z n 1 almost surely. 4. Consider independent and identically distributed random variables X 1 , X 2 , . . . such that P ( X k = 0) < 1. (a) (11 pts) Assuming that X k have finite mean, prove that P ( summationdisplay k X k converges ) = 0 . (b) (16 pts) Prove the same conclusion without the finite mean assumption. 5. (16 pts) A horizontal stick of length 1 is broken at a random point that is uniformly distributed. The right hand part is thrown away, and the left part is broken similarly (at a random point uniformly distributed in this part).part is broken similarly (at a random point uniformly distributed in this part)....
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 Spring '07
 RomanVershynin
 Probability

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