Lecture 18 - Transformation of RV

Lecture 18 - Transformation of RV - 1 TRANSFORMATION OF...

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1 TRANSFORMATION OF RANDOM VARIABLES (CONT.) Lecture 18 ORIE3500/5500 Summer2011 Li 1 Transformation of Random Variables (Cont.) 1.1 PDF One uses similar approach in order to find the density of a transformed random variable, but one has to proceed with care. If Y = T ( X ), X is continuous and T is one-one, that is increasing or decreasing, then f Y ( y ) = f X ( T - 1 ( y )) dT - 1 ( y ) dy Example Suppose X is a standard exponential random variable and set Y = 1 /X . This means that X has pdf f X ( x ) = e - x ,x > 0 . The transformation is T ( x ) = 1 /x and hence T - 1 ( y ) = 1 /y . This means that dT - 1 ( y ) dy = - 1 y 2 , and hence f Y ( y ) = f X (1 /y ) - 1 y 2 = 1 y 2 e - 1 /y ,y > 0 . This method can be extended to the case when the transformation T is not one-one. Then the T ( x ) = y has several roots, T - 1 1 ( y ) ,T - 1 2 ( y ) ,... . If Y = T ( X ), then Y will have density X i f X ( T - 1 i ( y )) dT - 1 i ( y ) dy Example Let X be a standard normal random variable, and Y = T ( X ), where T ( x ) = - x, x 0 2 x, x > 0 . 1
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This means that the range of Y is [0 , ) and for y > 0 T - 1 1 ( y ) = - y,T - 1 2 ( y ) = y/ 2 , for y = 0, T - 1 ( y ) = - y. This means for y > 0 dT - 1 1 ( y ) dy = - 1 , dT - 1 2 ( y ) dy = 1 2 , and for y = 0, dT - 1 ( y ) dy = - 1 . Hence f Y ( y ) is f Y ( y ) = f X ( - y ) | - 1 | + f X ( y 2 ) | 1 2 | = 1 2 π e - y 2 2 + 1 2 π e - y 2 8 1 2 = 1 2 π h e - y 2 / 2 + 1 2 e - y 2 / 8 i for y > 0 , and f Y (0) = 1 2 π . In summary, we could compute the pdf in three steps: 1. For y in the support of Y , get the inverse function T - 1 ( y ) (or T - 1 i ( y ) if T ( · ) is not a one to one mapping) 2. take the derivative(s) : dT - 1 ( y ) dy (or dT - 1 i ( y ) dy ) 3. Plug the absolute value of the derivative(s) and f X ( T - 1 ( y ))( f X ( T - 1 i ( y ))) in the formula and get the pdf of Y 1.2 PMF In case of discrete random variables one uses similar techniques to compute the pmf of the transformed random variable. If X is a discrete random variable and Y = T ( X ) for some transformation T , then p Y ( y j ) = P [ Y = y j ] = P [ X ∈ { x i : T ( x i ) = y j } ] = X i : T ( X i )= y j p X ( x i ) . If T is one-one then p Y ( y j ) = P [ X = T - 1 ( y j )] = p X ( T - 1 ( y j )) . 2
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Lecture 18 - Transformation of RV - 1 TRANSFORMATION OF...

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