where Px. r(, y) is the joint density of X and Y and Pr(y) is the density of Y. Using this conditional density, conditional expectations can be computed by Elf(X)IY]()=/ f()pxpr(cly)d.c (103) Here, we have emphasized the fact that the conditional expectation is a function of the values y of the random variable Y. If y is the g-algebra generated by y, then we write EL(XID=E(X)IY. Property(101) can be checked as follows E(Ef()I IF]=/ Ef(IX)](3)py(y)dy ∫(x)pxy(xly)py(y)d f(ar)PxY(a, y)d f(apx(r) EIf(XIF Example 3.2 Let X and w be independent random variables with densities Px(ar) and Pw(a), respectively. Let Y be a random variable defined by Y=X+w The joint density of X and Y is Px. r(a, y)=px()pw(y-z), and this defines a probability measure P(d x)=Px, r(a, y)d.c on the Borel subsets of Q2=RXR. The conditional density is given explicitly by Pxjy(aly) px(z)pw(y-c) Px(a)pw(y-a)dr which defines a conditional distribution T(y)(dr)=pxjy(aly)dr In the absence of measurement information, expectations of functions f(X) are eval- ated by (P, f)=E[]=/f()P(dx), in accordance with(99 ). Measurement of r provides information about X, allowing us to revise expectations using a conditional probability measure. Suppose we know that the values of Y occurred in a set F(FEy=a(Y)). The revised or conditional probability measure Is IF(up(a, y)dy P(F) 24where pX,Y (x, y) is the joint density of X and Y and pY (y) is the density of Y . Using this conditional density, conditional expectations can be computed by E[f(X)|Y ](y) = Z ∞ −∞ f(x)pX|Y (x|y)dx. (103) Here, we have emphasized the fact that the conditional expectation is a function of the values y of the random variable Y . If Y is the σ-algebra generated by Y , then we write E[f(X)|Y] = E[f(X)|Y ]. Property (101) can be checked as follows: E[Ef([X)|Y]IF ] = Z F Ef([X)|Y ](y)pY (y)dy = Z F Z ∞ −∞ f(x)pX|Y (x|y)pY (y)dxdy = Z F Z ∞ −∞ f(x)pX,Y (x, y)dxdy = Z F f(x)pX(x)dx = E[f(X)IF ]. Example 3.2 Let X and W be independent random variables with densities pX(x) and pW (w), respectively. Let Y be a random variable defined by Y = X + W. (104) The joint density of X and Y is pX,Y (x, y) = pX(x)pW (y−x), and this defines a probability measure P(dx) = pX,Y (x, y)dx on the Borel subsets of Ω = R×R. The conditional density is given explicitly by pX|Y (x|y) = pX(x)pW (y − x) Z pX(x)pW (y − x)dx , which defines a conditional distribution π(y)(dx) = pX|Y (x|y)dx. In the absence of measurement information, expectations of functions f(X) are eval￾uated by hP, fi = E[f] = Z f(x)P(dx), in accordance with (99). Measurement of Y provides information about X, allowing us to revise expectations using a conditional probability measure. Suppose we know that the values of Y occurred in a set F (F ∈ Y = σ(Y )). The revised or conditional probability measure is π(F)(dx) = Z IF (y)p(x, y)dy p(F) dx, (105) 24