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Unformatted text preview: 1 9. Two Functions of Two Random Variables In the spirit of the previous lecture, let us look at an immediate generalization: Suppose X and Y are two random variables with joint p.d.f Given two functions and define the new random variables How does one determine their joint p.d.f Obviously with in hand, the marginal p.d.fs and can be easily determined. (91) ). , ( y x f XY ). , ( ) , ( Y X h W Y X g Z = = ) , ( y x g ), , ( y x h (92) ? ) , ( w z f ZW ) , ( w z f ZW ) ( z f Z ) ( w f W PILLAI 2 The procedure is the same as that in (83). In fact for given z and w , where is the region in the xy plane such that the inequalities and are simultaneously satisfied. We illustrate this technique in the next example. ( ) ( ) ( ) ∫ ∫ ∈ = ∈ = ≤ ≤ = ≤ ≤ = w z D y x XY w z ZW dxdy y x f D Y X P w Y X h z Y X g P w W z Z P w z F , ) , ( , , ) , ( ) , ( ) , ( , ) , ( ) ( , ) ( ) , ( ξ ξ (93) w z D , z y x g ≤ ) , ( w y x h ≤ ) , ( x y w z D , Fig. 9.1 w z D , PILLAI 3 Example 9.1: Suppose X and Y are independent uniformly distributed random variables in the interval Define Determine Solution: Obviously both w and z vary in the interval Thus We must consider two cases: and since they give rise to different regions for (see Figs. 9.2 (a)(b)). ). , max( ), , min( Y X W Y X Z = = . or if , ) , ( < < = w z w z F ZW ). , ( w z f ZW ). , ( θ ). , ( θ (94) ( ) ( ) . ) , max( , ) , min( , ) , ( w Y X z Y X P w W z Z P w z F ZW ≤ ≤ = ≤ ≤ = (95) z w ≥ , z w < w z D , X Y w y = ) , ( w w ) , ( z z z w a ≥ ) ( X Y ) , ( w w ) , ( z z z w b < ) ( Fig. 9.2 PILLAI 4 For from Fig. 9.2 (a), the region is represented by the doubly shaded area. Thus and for from Fig. 9.2 (b), we obtain With we obtain Thus , , ) , ( ) , ( ) , ( ) , ( z w z z F z w F w z F w z F XY XY XY ZW ≥ − + = w z D , (96) , z w ≥ , z w < . , ) , ( ) , ( z w w w F w z F XY ZW < = (97) , ) ( ) ( ) , ( 2 θ θ θ xy y x y F x F y x F Y X XY = ⋅ = = (98) 2 2 2 (2 ) / , , ( , ) / , . ZW w z z z w F z w w w z θ θ θ θ − < < < = < < < (99) < < < = . otherwise , , , / 2 ) , ( 2 θ θ w z w z f ZW (910) PILLAI 5 From (910), we also obtain and If and are continuous and differentiable functions, then as in the case of one random variable (see (5 30)) it is possible to develop a formula to obtain the joint p.d.f directly. Towards this, consider the equations For a given point (z,w), equation (913) can have many solutions. Let us say , , 1 2 ) , ( ) ( θ θ θ θ < < − = = ∫ z z dw w z f z f z ZW Z (911) . ) , ( , ) , ( w y x h z y x g = = (913) . , 2 ) , ( ) ( 2 θ θ < < = = ∫ w w dz w z f w f w ZW W (912) ) , ( y x g ) , ( y x h ) , ( w z f ZW ), , ( , ), , ( ), , ( 2 2 1 1 n n y x y x y x " PILLAI 6 represent these multiple solutions such that (see Fig. 9.3) (914) . ) , ( , ) , ( w y x h z y x g i i i i = = Fig. 9.3 (b) x y 1 ∆ 2 ∆ i ∆ n ∆ • • • • ) , ( 1 1 y x ) , ( 2 2 y x ) , ( i i...
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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.
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