lect5 - 5. Functions of a Random Variable Let X be a r.v...

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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 = (5-1) ), ( y F Y ? ) ( y f Y ξ B Y ) ( ξ ) ( 1 B g )). ( ( ) ( 1 B g X P B Y P = (5-2) PILLAI
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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 x -axis 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 −∞ = = = ξ ξ ξ (5-3) ) ( ) ( 1 y g X ξ b aX + ) ( X g Y = 2 X | | X X ) ( | | x U X X e X log PILLAI
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3 Example 5.1: Solution: Suppose and On the other hand if then and hence b aX Y + = (5-4) . 0 > a () ( ) . ) ( ) ( ) ( ) ( = = + = = a b y F a b y X P y b aX P y Y P y F X Y ξ ξ ξ (5-5) . 1 ) ( = a b y f a y f X Y (5-6) , 0 < a , 1 ) ( ) ( ) ( ) ( = > = + = = a b y F a b y X P y b aX P y Y P y F X Y ξ ξ ξ (5-7) . 1 ) ( = a b y f a y f X Y (5-8) PILLAI
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4 From (5-6) and (5-8), 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 (5-9) . 2 X Y = () ( ) . ) ( ) ( ) ( 2 y X P y Y P y F Y = ξ (5-10) (5-11) , 0 < y { } , ) ( 2 φ ξ = y X . 0 , 0 ) ( < = y y F Y (5-12) , 0 > y } ) ( { } ) ( { 2 y X y Y = ξ ξ }. ) ( { 2 1 x X x < ξ 2 X Y = X y 2 x 1 x Fig. 5.1 PILLAI
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5 Hence By direct differentiation, we get If represents an even function, then (5-14) reduces to In particular if so that () . otherwise , 0 , 0 , ) ( ) ( 2 1 ) ( > + = y y f y f y y f X X Y (5-14) ) ( x f X ). ( 1 ) ( y U y f y y f X Y = (5-15) ), 1 , 0 ( N X . 0 ), ( ) ( ) ( ) ( ) ( ) ( 1 2 2 1 > = = < = y y F y F x F x F x X x P y F X X X X Y ξ (5-13) , 2 1 ) ( 2 / 2 x X e x f = π (5-16) PILLAI
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6 and substituting this into (5-14) or (5-15), we obtain the p.d.f of to be
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lect5 - 5. Functions of a Random Variable Let X be a r.v...

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