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Unformatted text preview: → 1-1) and diﬀerentiable function. 2a. Pr [ a < x < b ] = Pr [ g ( a ) < y < g ( b )] = R g ( b ) g ( a ) f y ( Y ) dY . 2b. Pr [ a < x < b ] = R b a f x ( X ) dX = R g ( b ) g ( a ) f x ( X = g-1 ( Y )) | dx/dy | dY . 2c. These are equal if f y ( Y ) = f x ( X = g-1 ( Y )) | dg-1 ( Y ) /dY | . EX1: y = 1 /x → X = g-1 ( Y ) = 1 /Y is decreasing and 1-1 for X,Y > 0. f y ( Y ) = | d (1 /Y ) dY | ‰ 1 2 if Y > 1 2 otherwise = ‰ 1 / (2 Y 2 ) if Y > 1 2 otherwise . EX2: Can’t use Jacobian since y = | x | not diﬀerentiable at x = 0....
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This note was uploaded on 07/22/2011 for the course EECS 370 taught by Professor Lehman during the Spring '10 term at University of Florida.
- Spring '10