This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: 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 (411), recall that the distribution function of X given an event B is (111) ( ) ( ) . ) ( ) ) ( (  ) ( )  ( B P B x X P B x X P B x F X ∩ ≤ = ≤ = ξ ξ PILLAI 2 Suppose, we let Substituting (112) into (111), we get where we have made use of (74). But using (328) and (77) we can rewrite (113) as To determine, the limiting case we can let and in (114). { } . ) ( 2 1 y Y y B ≤ < = ξ (113) (112) ( ) , ) ( ) ( ) , ( ) , ( ) ) ( ( ) ( , ) ( )  ( 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 (114) ),  ( y Y x F X = y y = 1 y y y ∆ + = 2 PILLAI 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 (117) with respect to x using (87), we get (115) (116) . ) ( ) , ( )  ( lim )  ( y f du y u f y y Y y x F y Y x F Y x XY X y X ∫ ∞ − → ∆ = ∆ + ≤ < = = (117) 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 = = (118) PILLAI 4 It is easy to see that the left side of (118) represents a valid probability density function. In fact and where we have made use of (714). From (119)  (1110), (118) 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 (118) and (1111), we have (119)  ( , ) (  ) ( ) XY X Y Y f x y f x Y y f y = = ≥ , 1 ) ( ) ( ) ( ) , ( )  (  = = = = ∫ ∫ ∞ + ∞ − ∞ + ∞ − y f y f y f dx y x f dx y Y x f Y Y Y XY Y X (1110) (1111) ).  ( )  (   y x f y Y x f Y X Y X = = , ) ( ) , ( )  (  y f y x f y x f Y XY Y X = (1112) PILLAI 5...
View
Full
Document
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
 prof

Click to edit the document details