lecture06

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

This preview shows pages 1–4. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
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 .

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

## 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.

### Page1 / 10

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

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online