ma503a-09-hm2

ma503a-09-hm2 - X with finite expectation, it holds that E...

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MA503a (FALL 2009) HOMEWORK #2 Due Wednesday, September 23, 2009 1) Recall that the Variance of a random variable X is defined by σ 2 X := E [ X - E ( X )] 2 . Show that (i) σ 2 X = E [ X 2 ] - ( E [ X ]) 2 ; (ii) σ 2 X = E { X ( X - 1) } + μ X - μ 2 X , where μ X = E [ X ]. Argue that μ 2 X E [ X 2 ], whenever both expectation exist. 2) Let g : [0 , ) 7→ [0 , ) be strictly increasing and nonnegative, and X be a random variable. Show that P {| X | ≥ a } ≤ E { g ( | X | ) } g ( a ) , for a > 0. 3) Let h : R 7→ [0 ] be a nonnegative (bounded) function. Show that for 0 a α , it holds that P { h ( X ) a } ≥ E { h ( X ) } - a α - a . 4) Using the definition of the conditional expectation, verify the following properties: (i) If X is G -measurable, then E { X |G} = X ; (ii) If G 1 ⊆ G 2 , then E { E { X |G 2 }|G 1 } = E { X |G 1 } ; (iii) If Y is G -measurable, then for any random variable
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Unformatted text preview: X with finite expectation, it holds that E { XY |G} = Y E { X |G} . 5) Let { B t : t ≥ } be a Brownian motion, and {F B t } t ≥ be the filtration generated by B . Using the definition of a Brownian motion to verify that B is indeed an {F B t }-martingale. 6) Suppose that { B t : t ≥ } is a standard Brownian motion. Show that for any constant c > 0, the scaled process W t := √ cB t/c , t ≥ 0 is also a standard Brownian motion. 7) Let σ , τ be two stopping times, show that both σ ∧ τ and σ ∨ τ are also stopping times. 1...
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This note was uploaded on 10/08/2010 for the course MA 503 taught by Professor Majin during the Fall '09 term at USC.

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