lectr11 - 11 Conditional Density Functions and Conditional...

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1 11. Conditional Density Functions and Conditional Expected Values As we have seen in section 4 conditional probability density functions are useful to update the information about an event based on the knowledge about some other related event (refer to example 4.7). In this section, we shall analyze the situation where the related event happens to be a random variable that is dependent on the one of interest. From (4-11), recall that the distribution function of X given an event B is (11-1) ( ) ( ) . ) ( ) ) ( ( | ) ( ) | ( B P B x X P B x X P B x F X = = ξ ξ PILLAI
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2 Suppose, we let Substituting (11-2) into (11-1), we get where we have made use of (7-4). But using (3-28) and (7-7) we can rewrite (11-3) as To determine, the limiting case we can let and in (11-4). { } . ) ( 2 1 y Y y B < = ξ (11-3) (11-2) ( ) , ) ( ) ( ) , ( ) , ( ) ) ( ( ) ( , ) ( ) | ( 1 2 1 2 2 1 2 1 2 1 y F y F y x F y x F y Y y P y Y y x X P y Y y x F Y Y XY XY X = < < = < ξ ξ ξ . ) ( ) , ( ) | ( 2 1 2 1 2 1 = < y y Y x y y XY X dv v f dudv v u f y Y y x F (11-4) ), | ( y Y x F X = y y = 1 y y y + = 2 PILLAI
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3 This gives and hence in the limit (To remind about the conditional nature on the left hand side, we shall use the subscript X | Y (instead of X ) there). Thus Differentiating (11-7) with respect to x using (8-7), we get (11-5) (11-6) . ) ( ) , ( ) | ( lim ) | ( 0 y f du y u f y y Y y x F y Y x F Y x XY X y X = + < = = (11-7) y y f y du y u f dv v f dudv v u f y y Y y x F Y x XY y y y Y x y y y XY X = + < + + ) ( ) , ( ) ( ) , ( ) | ( . ) ( ) , ( ) | ( | y f du y u f y Y x F Y x XY Y X = = . ) ( ) , ( ) | ( | y f y x f y Y x f Y XY Y X = = (11-8) PILLAI
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4 It is easy to see that the left side of (11-8) represents a valid probability density function. In fact and where we have made use of (7-14). From (11-9) - (11-10), (11-8) indeed represents a valid p.d.f, and we shall refer to it as the conditional p.d.f of the r.v X given Y = y . We may also write From (11-8) and (11-11), we have (11-9) 0 ) ( ) , ( ) | ( = = y f y x f y Y x f Y XY X , 1 ) ( ) ( ) ( ) , ( ) | ( | = = = = + + y f y f y f dx y x f dx y Y x f Y Y Y XY Y X (11-10) (11-11) ). | ( ) | ( | | y x f y Y x f Y X Y X = = , ) ( ) , ( ) | ( | y f y x f y x f Y XY Y X = (11-12) PILLAI
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5 and similarly If the r.vs X and Y are independent, then and (11-12) - (11-13) reduces to implying that the conditional p.d.fs coincide with their unconditional p.d.fs. This makes sense, since if X
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