Chapter 12 Conditional densities

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Chapter12Conditional OverviewDensity functions determine continuous distributions. If a continuous distri-bution is calculated conditionally on some information, then the density iscalled aconditional density. When the conditioning information involvesanother random variable with a continuous distribution, the Conditional den-sity can be calculated from the joint density for the two random a jointly continuous distribution with joint den-sityf(x,y). From Chapter 11, you know that the marginal distribution ofXis continuous with densityg(y) = f(x,y) Conditional distribution forYgivenX=xhas a ( Conditional ) density,which I will denote byh(y|X=x), or justh(y|x) if the conditioningvariable is unambiguous, for whichP{y Y y+ |X=x} h(y|X=x),for small > onX=xshould be almost the same as conditioning on theevent{x X x+ }for a very small >0.

is one of the few where a zero covariance (zero correlation) implies indepen-dence. The nal Example demonstrates yet another connection between Poisson processes and order statistics from a uniform distribution. The arguments make use of the obvious generalizations of joint densities and conditional densities to more than two dimensions. De nition.

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