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# lecture06 - 16.322 Stochastic Estimation and Control Fall...

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16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Lecture 6 Example: Sum of two independent random variables Z=X+Y b ( ) ( f z dz = P a < Z b ) z a ( = P a < X + Y b ) ( = P a X < Y b X ) ( ) ( = lim P x < X x + dx P a x < Y b x ) dx 0 x b x = lim f ( x dx f y dy ) ( ) dx 0 x y x a x b x ( ) y ( ) f x dx f y dy = x −∞ a x We can reach this same point by just integrating the joint probability density function for X and Y over the region for which the event is true. In the interior strip, the event a z b is true. < Page 1 of 10

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16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde b x ( < b ) = dx f , ( x y dy P a Z x y , ) −∞ a x b x ( ) f y dy dx f x = x y ( ) −∞ a x Let z x , = + y dz = dy b dx f x ( ) ( ) f z x dz = x y −∞ a b f x f z x dx dz ( ) y ( ) x = ∫ ∫ a −∞ This is true for all a,b . Therefore: ( ) = f x f z x dx f z x ( ) y ( ) z −∞ ( ) ( ) f y y f z y dy = x −∞ This result can readily be generalized to the sum of more independent random variables. Z = X 1 + X 2 + ... + X n f ( z ) = dx 1 dx 2 ... dx n 1 f ( x f ( x )... f ( x n 1 ) f x n ( z x x 2 ... x ) x 1 x n 1 1 n 1 z 1 ) x 2 2 −∞ −∞ −∞ Also, if W Y = X , for X,Y independent: ( ) = f x f w + x dx f w x ( ) y ( ) w −∞ ( ) ( ) f y y f y w dy = x −∞ Direct determination of the joint probability density of several functions of several random variables Suppose we have the joint probability density function of several random variables X,Y,Z , and we wish the joint density of several other random variables defined as functions X,Y,Z . = ± ( U u X Y Z ) , , V v X Y Z ) = ± ( , , = i ( W w X Y Z ) , , Page 2 of 10
16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde If f ( , , , , u v w ) can be found directly x y z ) is finite everywhere, the density f ( , , , , x y z u v w by the following steps: 1. Evaluate the Jacobian of the transformation from X,Y,Z to U,V,W . ± ( , , u x y z ) x ± ( , , v x y z ) ( , , J x y z ) = x ± ( , , w x y z ) x ± ( , , u x y z ) y ± ( , , v x y z ) y ± ( , , w x y z ) y ± ( , , u x y z ) z ± ( , , v x y z ) z ± ( , , w x y z ) z 2. For every value of u,v,w , solve the transformation equations for x,y,z . If there is more than one solution, get all of them. ± ( , , i ( , , u X Y Z ) = u x u v w ) ± ( , , ( , , v X Y Z ) = v y u v w ) i ± ( , , z u v w ) w X Y Z ) = w ( , , i 3. Then ( , , x y z ) f ( , , , , i i u v w ) = f x y z i , , u v w ( , , J x y z ) i i i k with x i ,y i ,z i given in terms of u,v,w . This approach can be applied to the determination of the density function for m variable which are defined to be functions of n variables ( n>m ) by adding some simple auxiliary variables such as x,y ,etc. to the list of m so as to total n variables.

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lecture06 - 16.322 Stochastic Estimation and Control Fall...

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