16. 322 Stochastic Estimation and Control, Fall 2004 Prof vander velde Conditioned on a value ofy: Let the event E be the event y<r sy+dy. E will be the value taken by y(like a measurement variable) lim f(x y)= f(x)f(ylx)dy f(u)f(yjudydu Note: y is not the variable of integration. f(xly= /(x)(y Ix) f(u)f(y judu This pre ovides a basis for the estimation of a parameter X based on prior information, f(x), and measurements y which are related to X. Example: y=x+nn=y-x parameterizing x f(yx)=f(y-x) We assume n is independent of x. f(ylx)is the noise distribution shifted by x f(n) fy×)=-fn(yx) Conditional Expectation The conditional expectation of a random variable or function of a random variable is the expectation calculated with the conditional distribution. E[g(x)1a= g(x)f(x[A)dx a useful relation involving conditional expectations E[8(x)=∑P(A4E[8(X)|4 if the Ai are mutually exclusive and exhaustive. Or if we let the events A, be the events y<Y≤y+dy,16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Conditioned on a value of Y: Let the event E be the event y Y y dy < ≤ + . E will be the value taken by y (like a measurement variable). ( | () ( | lim fxy ) = f xfyx)dy dy→0 ∞ f ( ) ( | u f y u)dydu ∫ −∞ Note: y is not the variable of integration. ( | () ( | fxy ) = fxfyx ) ∞ f ( ) ( | uf yu)du ∫ −∞ This provides a basis for the estimation of a parameter X based on prior information, f(x), and measurements y which are related to X. Example: y x n n y x = + = − parameterizing x f (| ( y x) = f y x − ) n We assume n is independent of x. f ( | y x)is the noise distribution shifted by x. Conditional Expectation The conditional expectation of a random variable or function of a random variable is the expectation calculated with the conditional distribution. ∞ E g x A [ ( )| ( ) ( | ) ] = g x f x A dx ∫ −∞ A useful relation involving conditional expectations is Egx [ ( )] = ∑PA E g X A ( ) i [ ( ) | i] i if the Ai are mutually exclusive and exhaustive. Or, if we let the events Ai be the events y Y y d <≤+ y , Page 8 of 9