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Unformatted text preview: 1 5. Functions of a Random Variable Let X be a r.v defined on the model and suppose g ( x ) is a function of the variable x . Define Is Y necessarily a r.v? If so what is its PDF pdf Clearly if Y is a r.v, then for every Borel set B , the set of for which must belong to F . Given that X is a r.v, this is assured if is also a Borel set, i.e., if g ( x ) is a Borel function. In that case if X is a r.v, so is Y , and for every Borel set B ), , , ( P F ). ( X g Y = (51) ), ( y F Y ? ) ( y f Y B Y ) ( ) ( 1 B g )). ( ( ) ( 1 B g X P B Y P = (52) PILLAI 2 In particular Thus the distribution function as well of the density function of Y can be determined in terms of that of X . To obtain the distribution function of Y , we must determine the Borel set on the xaxis such that for every given y , and the probability of that set. At this point, we shall consider some of the following functions to illustrate the technical details. ( ) ( ) . ] , ( ) ( )) ( ( ) ) ( ( ) ( 1 y g X P y X g P y Y P y F Y = = = (53) ) ( ) ( 1 y g X b aX + ) ( X g Y = 2 X   X X ) (   x U X X e X log PILLAI 3 Example 5.1: Solution: Suppose and On the other hand if then and hence b aX Y + = (54) . > a ( ) ( ) . ) ( ) ( ) ( ) ( = = + = = a b y F a b y X P y b aX P y Y P y F X Y (55) . 1 ) ( = a b y f a y f X Y (56) , < a ( ) ( ) , 1 ) ( ) ( ) ( ) ( = > = + = = a b y F a b y X P y b aX P y Y P y F X Y (57) . 1 ) ( = a b y f a y f X Y (58) PILLAI 4 From (56) and (58), we obtain (for all a ) Example 5.2: If then the event and hence For from Fig. 5.1, the event is equivalent to .   1 ) ( = a b y f a y f X Y (59) . 2 X Y = ( ) ( ) . ) ( ) ( ) ( 2 y X P y Y P y F Y = = (510) (511) , < y { } , ) ( 2 = y X . , ) ( < = y y F Y (512) , > y } ) ( { } ) ( { 2 y X y Y = }. ) ( { 2 1 x X x < 2 X Y = X y 2 x 1 x Fig. 5.1 PILLAI 5 Hence By direct differentiation, we get If represents an even function, then (514) reduces to In particular if so that ( ) . otherwise , , , ) ( ) ( 2 1 ) ( > + = y y f y f y y f X X Y (514) ) ( x f X ( ) ). ( 1 ) ( y U y f y y f X Y = (515) ), 1 , ( N X ( )...
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This note was uploaded on 10/16/2009 for the course EL el6303 taught by Professor Prof during the Spring '09 term at NYU Poly.
 Spring '09
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