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sta 2006 0809_lecturenotes_2

sta 2006 0809_lecturenotes_2 - Transformations Let X be a...

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Transformations Let X be a random variable with cdf F X ( x ). Let Y = g ( x ). For any set A , P ( Y A ) = P ( g ( X ) A ) = P ( X g - 1 ( A )) . 1. When X is discrete The pmf of Y is f Y ( y ) = P ( Y = y ) = x g - 1 ( y ) P ( X = x ) = x g - 1 ( y ) f X ( x ) for y ∈ Y , and f Y ( y ) = 0 for y not in Y . 2. When X is continuous The pdf of Y is f Y ( y ) = f x ( g - 1 ( y )) | d dy g - 1 ( y ) | y ∈ Y 0 otherwise Theorem : Let X have pdf f X ( x ), let Y = g ( X ), and define the sample space of X be X . Suppose there exists a partition, A 0 , A 1 , ..., A k , of X such that P ( X A 0 ) = 0 and f X ( x ) is continuous on each A i . Further, suppose there exist functions g 1 ( x ) , ..., g k ( x ), defined on A 1 , ..., A k , respectively, satisfying 1. g ( x ) = g i ( x ), for x A i , 2. g i ( x ), is monotone on A i , 3. the set Y = { y : y = g i ( x ) for some x A i } is the same for each i = 1 , ..., k , and 4. g - 1 i ( y ) has a continuous derivative on Y , for each i = 1 , ..., k . Then f Y ( y ) = k i =1 f x ( g - 1 i ( y )) | d dy g - 1 i ( y ) | y ∈ Y 0 otherwise 14
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Examples 1. Let X U (0 , 1). Find the distribution of Y = X n . 2. Let X have standard normal distribution. Find the pdf of Y = X 2 . 15
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Conditional Distributions Definition : Let ( X, Y ) be a discrete bivariate random vector with joint pmf f ( x, y ) and marginal pmfs f X ( x ) and f Y ( y ). For any x such that P ( X = x ) = f X ( x ) > 0, the conditional pmf of Y given X = x is the function of y denoted by f ( y | x ) and defined by f ( y | x ) = P ( Y = y | X = x ) = f ( x, y ) f X ( x ) . Definition : Let ( X, Y ) be a continuous bivariate random vector with joint pdf f ( x, y ) and marginal pdfs f X ( x ) and f Y ( y ). For any x such that P ( X = x ) = f X ( x ) > 0, the conditional pdf of Y given X = x is the function of y denoted by f ( y | x ) and defined by f ( y | x ) = f ( x, y ) f X ( x ) .
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