lect8a

# lect8a - 8 One Function of Two Random Variables Given two...

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1 8. One Function of Two Random Variables Given two random variables X and Y and a function g ( x , y ), we form a new random variable Z as Given the joint p.d.f how does one obtain the p.d.f of Z ? Problems of this type are of interest from a practical standpoint. For example, a receiver output signal usually consists of the desired signal buried in noise, and the above formulation in that case reduces to Z = X + Y . ). , ( Y X g Z = ), , ( y x f XY ), ( z f Z (8-1) PILLAI

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2 It is important to know the statistics of the incoming signal for proper receiver design. In this context, we shall analyze problems of the following type: Referring back to (8-1), to start with ) , ( Y X g Z = Y X + ) / ( tan 1 Y X Y X XY Y X / ) , max( Y X ) , min( Y X 2 2 Y X + ( ) ( ) [ ] ∫ ∫ = = = = z D y x XY z Z dxdy y x f D Y X P z Y X g P z Z P z F , , ) , ( ) , ( ) , ( ) ( ) ( ξ (8-2) (8-3) PILLAI
3 where in the XY plane represents the region such that is satisfied. Note that need not be simply connected (Fig. 8.1). From (8-3), to determine it is enough to find the region for every z , and then evaluate the integral there. We shall illustrate this method through various examples. z D z y x g ) , ( ) ( z F Z z D z D X Y z D z D Fig. 8.1 PILLAI

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4 Example 8.1: Z = X + Y. Find Solution: since the region of the xy plane where is the shaded area in Fig. 8.2 to the left of the line Integrating over the horizontal strip along the x -axis first (inner integral) followed by sliding that strip along the y -axis from to (outer integral) we cover the entire shaded area. ( ) + −∞ = −∞ = = + = , ) , ( ) ( y y z x XY Z dxdy y x f z Y X P z F (8-4) z D z y x + . z y x = + + y z x = x y Fig. 8.2 ). ( z f Z PILLAI
5 We can find by differentiating directly. In this context, it is useful to recall the differentiation rule in (7- 15) - (7-16) due to Leibnitz. Suppose Then Using (8-6) in (8-4) we get Alternatively, the integration in (8-4) can be carried out first along the y -axis followed by the x -axis as in Fig. 8.3. ) ( z F Z ) ( z f Z = ) ( ) ( . ) , ( ) ( z b z a dx z x h z H (8-5) ( ) ( ) + = ) ( ) ( . ) , ( ), ( ) ( ), ( ) ( ) ( z b z a dx z z x h z z a h dz z da z z b h dz z db dz z dH (8-6) ( , ) ( ) ( , ) ( , ) 0 ( , ) . z y z y XY Z XY XY XY f x y f x y dx dy f z y y d z z f z y y dy +∞ +∞ −∞ −∞ −∞ −∞ +∞ −∞ = = + = (8-7) PILLAI f z y

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6 In that case and differentiation of (8-8) gives + −∞ = −∞ = = , ) , ( ) ( x x z y XY Z dxdy y x f z F (8-8) + −∞ = + −∞ = −∞ = = = = . ) , ( ) , ( ) ( ) ( x XY x x z y XY Z Z dx x z x f dx dy y x f z dz z dF z f (8-9) If X and Y are independent, then and inserting (8-10) into (8-8) and (8-9), we get ) ( ) ( ) , ( y f x f y x f Y X XY = . ) ( ) ( ) ( ) ( ) ( + −∞ = + −∞ = = = x Y X y Y X Z dx x z f x f dy y f y z f z f (8-10) (8-11) x z y = x y Fig. 8.3 PILLAI
7 The above integral is the standard convolution of the functions and expressed two different ways. We thus reach the following conclusion: If two r.vs are independent, then the density of their sum equals the convolution of their density functions.

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