hw1 - w = ± 2 π ² 1 2 σ-3 w 2 exp ±-w 2 2 σ 2 ²...

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AMS316 Homework #1 (due Sept 14, 2011) This problem set is used as a self-test of your background on AMS311 and AMS315, it doesn’t represent the format or level of homework in AMS316. The grade of homework 1 will be not counted for final grade. 1. Let the random variable X and Y have joint distribution P ( X = a,Y = 0) = P ( X = 0 ,Y = a ) = P ( X = - a,Y = 0) = P ( X = 0 ,Y = - a ) = 1 / 4 . Show that X - Y and X + Y are independent. 2. The beta function B ( a,b ) is given by B ( a,b ) = R 1 0 v a - 1 (1 - v ) b - 1 dv ; a > 0 ,b > 0. The beta distribution has density f ( x ) = 1 B ( a,b ) x a - 1 (1 - x ) b - 1 , for 0 < x < 1 . If X has the beta distribution, show that E ( X ) = B ( a +1 ,b ) /B ( a,b ). What is Var( X )? 3. A molecule M has velocity v = ( x,y,z ) in Cartesian coordinates. Suppose that x , y and z have joint density: f ( x,y,z ) = (2 πσ 2 ) - 3 / 2 exp ± - 1 2 σ 2 ( x 2 + y 2 + z 2 ) ² . Show that the density of the magnitude | v | = p x 2 + y 2 + z 2 of v is f
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Unformatted text preview: ( w ) = ± 2 π ² 1 / 2 σ-3 w 2 exp ±-w 2 2 σ 2 ² , w > . (Hint: Argue that x,y,z are independent first, and then notice that v 2 /σ 2 follows a χ 2 distribution). 4. Consider a continuous random variable X with density function f ( x ), mean μ and variance σ 2 . Show that σ 2 ≥ Z | x-μ |≥ ± ( x-μ ) 2 f ( x ) dx ≥ ± 2 Z | x-μ |≥ ± f ( x ) dx, and furthermore, P ( | X-μ | ≥ ± ) ≤ σ 2 ± 2 . 5. Let X 1 ,...,X n are independent and identically distributed random variables with mean μ and variance σ 2 . Apply the result in Problem 4 for X = ∑ n i =1 X i ³ n to show the law of large number....
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