so that(R, B(R), Px) is a probability space. If X has a density px(r), expectations of unctions f(X)(e.g. moments) can be calculated via EU(X)=/f(e)Px(da f(apx() Note that the cumulative distribution function is given by Fx()=Px(oo, a]),and the density px(z )=dFx(a)d (when it exists) Let y be a random variable, and y= a(r be the a-algebra generated by Y;i.e the smallest o-algebra with respect to which Y is measurable. In general y C. If Z is a random variable that is also J-measurable, then there is a function gz such that Z(w)=gz(Y(w)). Thus, Z is a function of the values of Y For 0<p<oo, the set LP(Q, F, P) is the vector space of complex-valued random variables X such that EIlXP is finite. It is a Banach space with respect to the norm X p=EXP / P. The case p=2 is of special interest, since L(Q2, F, P)is a Hilbert space with inner product Y=EIXYI For p=oo, the space of essentially bounded random variables L(Q2, F, P)is a Banach space with norm‖x‖s= ess. sup|X(u) Example 3.1 Consider the classical probability space(Q2, F, P), where Q=(1, 21, F 10, 0, 1, 2)1, and P=(p1, p2). A random variable X has the form X=(r1, r2), where 1,12 E C, and in this example, the spaces L(Q2, F, P)(equal to L2(Q, F, P)as a set consist of all such random variables. The expected value of X is given by E[X]=/ X()P(dw)=T1P1 +x2p (100) 3.1.2 Conditional Expectations Let y C F be a sub-o-algebra. The conditional expectation e(X)I of f(X) given y is the unique J-measurable function such that EI(XIF=EEf(X)IIF] for all FE J (101) Here, IF is the indicator function of the set F defined by IF(w)=l if w E F, and IF(w)=0 otherwise This definition may seem abstract, and so we attempt to clarify it by describing what happens in the language of elementary probability and give examples Recall from elementary probability the notion of conditional density pxjy(aly) of a random variable X given another random variable y. It is given by Pxlr(alv)=Px,r(z, y) (102)so that (R, B(R), PX) is a probability space. If X has a density pX(x), expectations of functions f(X) (e.g. moments) can be calculated via E[f(X)] = Z R f(x)PX(dx) = Z ∞ −∞ f(x)pX(x)dx. (99) Note that the cumulative distribution function is given by FX(x) = PX((−∞, x]), and the density pX(x) = dFX(x)/dx (when it exists). Let Y be a random variable, and Y = σ(Y ) be the σ-algebra generated by Y ; i.e. the smallest σ-algebra with respect to which Y is measurable. In general Y ⊂ F. If Z is a random variable that is also Y-measurable, then there is a function gZ such that Z(ω) = gZ(Y (ω)). Thus, Z is a function of the values of Y . For 0 < p < ∞, the set L p (Ω, F, P) is the vector space of complex-valued random variables X such that E[|X| p ] is finite. It is a Banach space with respect to the norm k X kp= E[|X| p ] 1/p. The case p = 2 is of special interest, since L 2 (Ω, F, P) is a Hilbert space with inner product hX, Y i = E[X ∗Y ]. For p = ∞, the space of essentially bounded random variables L ∞(Ω, F, P) is a Banach space with norm k X k∞= ess.supω |X(ω)|. Example 3.1 Consider the classical probability space (Ω, F, P), where Ω = {1, 2}, F = {∅, Ω, {1}, {2}}, and P = (p1, p2). A random variable X has the form X = (x1, x2), where x1, x2 ∈ C, and in this example, the spaces L ∞(Ω, F, P) (equal to L 2 (Ω, F, P) as a set) consist of all such random variables. The expected value of X is given by E[X] = Z Ω X(ω)P(dω) = x1p1 + x2p2. (100) 3.1.2 Conditional Expectations Let Y ⊂ F be a sub-σ-algebra. The conditional expectation E[f(X)|Y] of f(X) given Y is the unique Y-measurable function such that E[f(X)IF ] = E[Ef(X)|Y]IF ] for all F ∈ Y. (101) Here, IF is the indicator function of the set F defined by IF (ω) = 1 if ω ∈ F, and IF (ω) = 0 otherwise. This definition may seem abstract, and so we attempt to clarify it by describing what happens in the language of elementary probability and give examples. Recall from elementary probability the notion of conditional density pX|Y (x|y) of a random variable X given another random variable Y . It is given by pX|Y (x|y) = pX,Y (x, y) pY (y) (102) 23