hw5 - X and Y , one has E | X-Y | ≤ 1 2 . 5. Independence...

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Homework 5 Probability Theory (MATH 235A, Fall 2007) 1. Random series. Let X 1 ,X 2 ,... be non-negative random variables. Prove the equality E X n =1 X n · = X n =1 E X n . Explain in what sense the equality holds. 2. Sharpness of Chebyshev’s inequality. For every t 1, construct a random variable X with mean μ and variance σ 2 , and such that Chebyshev’s inequality becomes an equality: P ( | X - μ | ≥ ) = 1 t 2 . 3. Averaging a density Prove that, for any random variable X and a > 0, one has Z R P ( x < X < x + a ) dx = a. 4. Uniform distribution Let X and Y be uniformly distributed random variables on [0 , 1]. Show that, whatever the dependence between
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Unformatted text preview: X and Y , one has E | X-Y | ≤ 1 2 . 5. Independence Let X 1 ,...,X n be random variables such that P ( ( X 1 ,...,X n ) ∈ A ) = Z A f ( x ) dx for every Borel set A in R n . Assume that f : R n → R can be factored as f ( x 1 ,...,x n ) = f 1 ( x 1 ) ··· f n ( x n ) for some non-negative measurable functions f k ; R → R . Prove that X 1 ,...,X n are independent. Note that f k are not assumed to be density functions. (How-ever, if you feel a need of continuity, you may assume that f k are continuous)....
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This note was uploaded on 01/25/2011 for the course STAT 235a at Stanford.

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