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Unformatted text preview: Todays outline: pp 4755 Examples of variable transformation Onetoone transformations Theorems Example 1: Exponential distn [similar to HW 1.53, 1.55] We are given a random variable X with pdf f X ( x ) = exp( x ) , x > . We say that X Exp ( ) is exponentially distributed with parameter (also called rate) . What is the cdf of X ? What is the pdf of Y = 2 X ? How about Z = X ? P ( Y < y ) = P (2 X < y ) = P ( X < y/ 2) = Z y/ 2 f X ( x ) dx = 1 exp( y/ 2) , P ( Z < z ) = 1 exp( z ) . Differentiating, the pdf of Z becomes g ( z ) = exp( z ) So Y and Z are also exponentially distributed, with parameters / 2 and 1, respectively. Example 2: Exponential distribution As before, X with pdf f X ( x ) = exp( x ) , x > . Take Y = X 2 . What is the cdf of Y ? P ( Y < y ) = P ( X 2 < y ) = P ( X < y ) = 1 exp( p ( y )) . The pdf becomes f Y ( y ) = exp( y ) 2 y ....
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This note was uploaded on 01/16/2011 for the course STAT 213 taught by Professor Ioannam during the Fall '09 term at Duke.
 Fall '09
 IoannaM

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