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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 () ( fz d z = Pa < Z b ) z a ( = Pa < X + Y b ) ( = X < Y b X ) ( ) ( = lim Px < X x + d xPa x < Y b x ) dx 0 x bx = lim f ( x dx f y dy ) ( ) dx 0 x y x ax y ( ) f x dx f y dy = 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 az b is true. < Page 1 of 10
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16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde bx ( <≤ b ) = dx f , ( x y dy P a Z ∫∫ xy , ) −∞ ax () f y dy dx f x = x y ( ) −∞ ax Let zx , =+ y d z = 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 z x d x f z x ( ) y ( ) z −∞ () ( ) f y yf yd y = 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 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 WY =− X , for X,Y independent: = w + x d x f w x ( ) y ( ) w −∞ ) f y yf y wd y = 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 . = ± ( Uu X Y Z ) ,, Vv X Y ) = ± ( = i ( Ww X ) Page 2 of 10
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16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde If f (, , ,, uvw ) can be found directly xyz ) is finite everywhere, the density f (,, ,, uvw by the following steps: 1. Evaluate the Jacobian of the transformation from X,Y,Z to U,V,W . ± , uxyz ) x ± , vxyz ) , Jxyz ) = x ± , wxyz ) x ± , ) y ± , ) y ± , ) y ± , ) z ± , ) 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 uXY Z ) = u x ) ± ( (,, vXY ) = v y ) i ± ( z ) wXY ) = w i 3. Then , xy ) f i i u v w ) = f xyz i Jx ) 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. Then apply this procedure and finally integrate out the unwanted auxiliary variables. Example: Product U=XY To illustrate this procedure, suppose we are given f xy (, ) and wish to find , the probability density function for the product U = XY .
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This note was uploaded on 11/07/2011 for the course AERO 16.322 taught by Professor Wallacevandervelde during the Fall '04 term at MIT.

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

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