We will also discuss conditional variance. An important concept here is that we interpret the conditional expectation as a random variable. Conditional Expectation as a Function of a Random Variable: Remember that the conditional expectation of XX given that Y=yY=y is given by ...
conditional expectation(条件期望讲义)A Conditional expectation A.1Review of conditional densities,expectations We start with the continuous case.This is sections6.6and6.8in the book.Let X,Y be continuous random variables.We defined the conditional density of X given Y to be f X,Y(x,y)f ...
The conditional expectation (or conditional expected value, or conditional mean) is the expected value of a random variable, computed with respect to a conditional probability distribution. A pragmatic approachAs in the case of the expected value, a completely rigorous definition of the conditional ...
The discrete conditional expectation E(Y | X) of a numerical random variable Y: (Ω, , P) (R,) given a random variable X: (Ω, , P) (Ω' X , ' X ) is now defined as that random variable on (Ω, , P) whose values are identical to the conditional expectation values E(Y |...
Mathematical expectation is viewed as a probability weighted average over the set of all possible values of a random variable. The idea that knowledge of the occurrence of an event C modifies the possible outcomes—restricts them to set C —is utilized to modify the manner in which probabilities...
NowwedefineconditionalexpectationE[X|Y]asarandomvariablewhichtakesvalueE[XY=| y i ]withprobabilityp i ,whereE[X|Y=y i ]shouldbeunderstoodasexpectationofXconditioned ontheevent{ω∈Ω:Y(ω)=y i }. Itturnsoutthatonecandefineconditionalexpectationwithrespecttoaσfield.Thisnotion ...
Theminimum mean squared error (MMSE)estimate of the random variableXX, given that we have observedY=yY=y, is given by x^M=E[X|Y=y].x^M=E[X|Y=y]. Example XX XX Y=yY=y Solution First we need to find the posterior density,fX|Y(x|y). We have ...
The conditional expectation of a random function g(x, y) is computed as 随机函数 g(x,y) 的条件期望的计算方式为 E[g|x]=∫g(x,y)p(y|x)dy=∫g(x,y)p(x,y)dy∫p(x,y)dy. These definitions are useful for multidimensional Monte Carlo computations. 这些定义对于多维蒙特卡罗计算很...
As with the initial study of random variables in the beginning of this chapter, it is convenient to start with the notion of a conditional CDF. DEFINITION 3.4: The conditional cumulative distribution function of a random variable, X, conditioned on the event A having occurred is (3.35)FX|A(...
whereEBZdenotes the conditional expectation of a random variableZgiven ℬ. In contrast to the ordinary concept of variance, conditional variance ofXgiven ℬ is defined asVarBX=CovB(X,X). In the following, we present some concepts of uniform integrability. ...