hw1 - ( w ) = 2 1 / 2 -3 w 2 exp -w 2 2 2 , w >...

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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 1 / 2 -3 w 2 exp -w 2 2 2 , w &gt; . (Hint: Argue that x,y,z are independent rst, and then notice that v 2 / 2 follows a 2 distribution). 4. Consider a continuous random variable X with density function f ( x ), mean and variance 2 . Show that 2 Z | x- | ( x- ) 2 f ( x ) dx 2 Z | x- | f ( x ) dx, and furthermore, P ( | X- | ) 2 2 . 5. Let X 1 ,...,X n are independent and identically distributed random variables with mean and variance 2 . Apply the result in Problem 4 for X = n i =1 X i n to show the law of large number....
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