Past exam of the options I course of the University of Oxford / 2012 / M / M2 / b / iv

For $0<x<1$, the conditional density is obtained by dividing the joint density by the positive marginal density:Consequentlywhich integrates to one. Its conditional expectation isThis uses the usual continuous conditional-density version, defined by the density ratio at a point where $f_X$ is positive; conditioning does not require the event $X=1/2$ itself to have positive probability.