conditional density and conditional expectation suppose and are both discrete or both continuouswith joint pdf and suppose fx ( thenthe conditional pdf of given is given by fy| ( xf (xyfx (xfor all conditional pdf ( in the discrete casethe formula is direct translation of ( )with fy| ( xp[ xin the continuous casea similar interpretation in terms of densities can be usedseefor example[ page the corresponding distribution is called the conditional distribution of given note that ( implies that (xyfx (xfy ( xconditional distribution this is useful when the marginal and conditional pdfs are givenrather than the joint one more generallyfor the -dimensional case we have ( xn fx ( fx ( fxn ,xn- (xn xn- )( which is in essence rephrasing of the product rule ( in terms of probability densities as conditional pdf has all the properties of an ordinary pdfwe may define expectations with respect to it the conditional expectation of random variable given is defined as discrete casey fy| ( xr [ ( fy| ( xdy continuous case conditional expectation note that [ xis function of the corresponding random variable is written as [ xa similar formalism can be used when conditioning on sequence of random variables xn the conditional expectation has similar properties to the ordinary expectation other useful properties (seefor example[ ]are tower propertyif ey existsthen [ xey ( taking out what is knownif ey existsthen [xy xxe[ xc functions of random variables let [ xn ]be column vector in rn and an matrix the mapping zwith axis linear transformationas discussed in section now consider random vector [ xn ]and let :ax then is random vector in rm the following theorem details how the distribution of is related to that of