16.322 Stochastic Estimation and Control, Fall 2004 Prof vander velde E[8(x]=E[g(x)ly]/(y)dy The conditional expectation of a random variable, Exlyl, is often taken as the estimate of the random variable after incorporating the effect of a measured event or related random variable f(x EI,En): State of knowledge about x given observations El,En f(xiyi,,y,n): State of knowledge about x given measurements y,,,'n Probability Distribution of Functions of Random variables The probability distribution and density for functions of random variables can be calculated from the distributions of the variables themselves Z In general, if Z=g(X, y) and we want f(=),we can use F(二)=P(z≤=) =P[g(X,)≤ =」d,(xy to get the distribution function for all values of (x)=F(=) In particular cases, there may be easier ways to do this, but this is the general ocedure Page 9 of 916.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde ∞ Egx [ ( )] = Egx [ ( ) | y f y dy ] ( ) ∫ −∞ The conditional expectation of a random variable, E[x y| ], is often taken as the estimate of the random variable after incorporating the effect of a measured event or related random variable. f (x E| 1 ,...,En ) : State of knowledge about x given observations E1,…,En f (x y| 1 ,..., yn ) : State of knowledge about x given measurements y1,…,yn Probability Distribution of Functions of Random Variables The probability distribution and density for functions of random variables can be calculated from the distributions of the variables themselves. In general, if Z = gXY (, ) and we want f ( )z , we can use z Fz ( ) = PZ ( ≤ z) z = PgXY [ ( , ) ≤ z] dx dy f x y = ( , x y ) ∫ ∫ , to get the distribution function for all values of z. dF z z f z( ) = () z dz In particular cases, there may be easier ways to do this, but this is the general procedure. Page 9 of 9