hw4 - | E X | ≤ r E X 2 5 Let X be a non-negative random...

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Math 280A, Fall 2011 Homework 4 1. Problem 16 (page 116) 2. Problem 17 (page 116) 3. (Schef´ e’s Lemma) Let X, X 1 , X 2 , . . . be non-negative integrable random variables. Suppose that X n X a.s. and E [ X n ] E [ X ] as n → ∞ . Show that E | X - X n | → 0 as n → ∞ . [Hints: IF b is a real number then | b | = 2 b + - b . Use this with b = X ( ω ) - X n ( ω ), noting that ( X - X n ) + X . Apply DCT.] 4. Let X be a random variable with E [ X 2 ] < . [ X is then integrable, because | X | ≤ 1 + X 2 .] Consider the Function v ( t ) := E b ( X - t ) 2 B , t R . (a) Show that v takes on its minimum value at t = E [ X ]. [The minimum value is thereFore Var[ X ]. (b) ±rom (a) deduce the inequality
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Unformatted text preview: | E [ X ] | ≤ r E [ X 2 ] . 5. Let X be a non-negative random variable. Show that ∞ s n =1 P [ X > n ] ≤ E [ X ] ≤ ∞ s n =0 P [ X > n ] . (A more precise version oF this Fact will emerge From Tonelli’s theorem, in due course.) [Hint: ±or the leFt-hand inequality, begin by showing that X ( ω ) ≥ ∑ ∞ n =1 1 { X>n } ( ω ) For each ω ∈ Ω. Deal with the right-hand inequality in similar Fashion.]...
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This note was uploaded on 12/11/2011 for the course MATH 280a taught by Professor Driver,b during the Fall '08 term at UCSD.

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