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Unformatted text preview: ( w ) = 2 1 / 2 3 w 2 exp w 2 2 2 , w > . (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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 Fall '09

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