16.322 Stochastic Estimation and Control, Fall 2004 Prof vander velde If Xi and X are independent, their covariance is zero If the covariance between two variables is zero, they are said to be uncorrelated This does not, in general, imply that they are independent Conditional distribution Sometimes the notion of a conditional distribution is important. If we wish to confine our attention to the subject of cases in which an event e occurs, we would define the conditional probability distribution function FE(x)=f(x E)=P(X<xE) P(XSx, E) P(E) and the conditional probability density function f(x)=f(x e)= Application of Bayes'Rule Bayes theorem can be written for the distribution of a random variable conditioned on the occurrence of an event e or the observation of a certain value of another random variable Conditioned n event e Original form: P(A IE)=P(A)P(EIA) ∑P(A)P(E|4) Let the events a→ the events x<X≤x+d Note that for different values x as dx>0 these events are mutually exclusive and collectively exhaustive. lim f(x E)dx f(dxp(ex) f(uP(eju)du f(xP(Elx) f(uP(eJudu16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde If Xi and Xj are independent, their covariance is zero. If the covariance between two variables is zero, they are said to be uncorrelated. This does not, in general, imply that they are independent. Conditional Distribution Sometimes the notion of a conditional distribution is important. If we wish to confine our attention to the subject of cases in which an event E occurs, we would define the conditional probability distribution function: Fx ( ) = F x E ( | ( | ) = PX ≤ x E ) E P X( , ≤ xE ) = PE ( ) and the conditional probability density function: dF x E ( ) = f x E ) = ( ) fx ( | E dx Application of Bayes’ Rule Bayes’ theorem can be written for the distribution of a random variable conditioned on the occurrence of an event E or the observation of a certain value of another random variable. Conditioned on an event E: ( | ( )( | Original form: k k P A E ) = P A P E A ) k ∑P A P E A ( ) ( | ) i i i Let the events A x i → the events x < X ≤ + dx Note that for different values x as dx → 0 these events are mutually exclusive and collectively exhaustive. ( | ) () ( | lim f x E dx = ∞ f x dxP E x ) dx→0 f ( ) ( | ) u P E u du ∫ −∞ (| ()( | f x E ) = f x P E x ) ∞ f ( ) ( | ) u P E u du ∫ −∞ Page 6 of 9