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Unformatted text preview: Functions of Random Variables Enrique CamposN anez The George Washington University ApSc 116 1 / 32 Table of Contents Functions of a Single Random Variable Functions of Several Random Variables Sum of Random Variables 2 / 32 Functions of a Single Random Variable Discrete Random Variable Let X be a discrete r.v. and Y r ( X ) be a function of X , i.e., s S , we define Y ( s ) = r ( X ( s )). Then Y is also a discrete r.v. (why?) The p.f. g of Y is obtained as follows: g ( y ) = Pr ( Y = y ) = Pr ( r ( X ) = y ) = X x : r ( x )= y f ( x ) . Example Let X be a random variable with p.f. f ( x ) = Pr ( X = x ) = ( 1 / 9 if x { 4 , 3 , . . . , 4 } , otherwise. Now, we define Y X 2 . Then, the p.f. g of Y is given by g ( y ) = Pr ( Y = y ) = Pr ( r ( X ) = y ) = 2 / 9 if y { 1 , 4 , 9 , 16 } , 1 / 9 if y = 0, and otherwise. 3 / 32 Functions of a Single Random Variable Continuous Random Variable Let X be a continuous r.v. and let Y = r ( X ). Let f ( x ) be the p.d.f. of X . Is Y a continuous r.v.? Let G ( y ) be the d.f. of Y ; we obtain it as follows: G ( y ) Pr ( Y y ) = Pr ( r ( X ) y ) = Z { x : r ( x ) y } f ( x ) dx If Y has a continuous probability distribution, we obtain its p.d.f. g ( y ) from g ( y ) = G ( y ) y Example Let X be a r.v. with the exponential distribution, i.e. with p.d.f. f ( x ) = e x for x 0; let Y = 1 / X . Then G ( y ) = e / y for y , g ( y ) = y 2 e / y for y 0. 4 / 32 Direct Derivation of the Probability Density Function I Suppose Y = r ( X ) and both X and Y are continuous r.v.s Suppose, also, that Pr ( a < X < b ) and r ( x ) is a strictly increasing function over the interval ( a , b ), (e.g., ln x ). Note that a < X < b r ( a ) < Y < r ( b ) . With the additional assumption that r ( x ) is a continuous function of x over ( a , b ), it follows that y ( , ) there is a unique x ( a , b ), which we call s ( y ), such that r ( x ) = y . Then s ( y ) is the inverse function of r ( x ) so that y = r ( x ) iff x = s ( y ). 5 / 32 Direct Derivation of the Probability Density Function II If, s ( y ) is continuous and strictly increasing over the interval ( , ). Hence, for y ( , ) we have G ( y ) = Pr ( Y y ) = Pr ( r ( X ) y ) = Pr ( X s ( y )) = F ( s ( y )) . If s ( y ) is a differentiable function over ( , ), then Y is a continuous r.v. and its p.d.f. g ( y ) for < y < is found as g ( y ) = G ( y ) y = F ( s ( y )) y = f ( s ( y )) s ( y ) y , by the chain rule. 6 / 32 Direct Derivation of the Probability Density Function III A similar analysis holds if r ( x ) is a continuous and strictly decreasing function over ( a , b ). Then = r ( b ), = r ( a ), and the inverse function s ( y ) is continuous and strictly decreasing over ( , ). Then we get G ( y ) = Pr ( Y y ) = Pr ( r ( X ) y ) = Pr ( X s ( y )) = 1...
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This note was uploaded on 01/02/2010 for the course APSC 116 taught by Professor Tommazzuchi during the Fall '08 term at GWU.
 Fall '08
 TomMazzuchi

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