论文标题
关于随机变量的有条件期望的定义定义版本
Pointwise defined version of conditional expectation with respect to a random variable
论文作者
论文摘要
在一个随机变量给出的单数事件中的条件通常很感兴趣，例如对于连续的随机变量$ y $，$ \ {y = y \} $。相对于此事件的条件度量通常被推导为相对于生成Sigma代数的随机变量的条件期望的特殊情况。后者的存在通常通过非构造措施理论论证证明，该论点几乎唯一的定义数量。 In particular, the quantity $\mathbb E[f|Y]$ is initially only defined almost everywhere and conditioning on $Y=y$ corresponds to evaluating $\mathbb E[f|Y=y] = \mathbb E[f|Y]{Y=y}$, which is not meaningful because of $\mathbb E[f|Y]$ not being well-defined on such singular sets.引入常规条件分布也不会解决此问题。另一方面，可以表明，天真计算的条件密度$ f_ {z | y = y}（z）$（由关节和边际密度的比例给出）是条件分布的一种版本，即$ \ sathbb e [\ y \ y {z \ in b \ y = y = y = y]实际上，以$ y $的价格评估。数学理论之间的不匹配（生成无法从中产生我们需要的对象），而通过条件密度进行实际计算是一个不幸的事实。此外，经典方法不允许对$ \ mathbb e [f | y = y] $的条件期望的有条件期望定义，仅是条件分布的$ \ mathbb e [\ {z \ in b \ \} | y = y] $。我们提出了一种（据作者所知）鲜为人知的方法，可以通过使用Lebesgue-Besicovich引理获得有条件期望的定义定义版本，而无需在通常的派生中需要其他拓扑参数。
It is often of interest to condition on a singular event given by a random variable, e.g. $\{Y=y\}$ for a continuous random variable $Y$. Conditional measures with respect to this event are usually derived as a special case of the conditional expectation with respect to the random variables generating sigma algebra. The existence of the latter is usually proven via a non-constructive measure-theoretic argument which yields an only almost-everywhere defined quantity. In particular, the quantity $\mathbb E[f|Y]$ is initially only defined almost everywhere and conditioning on $Y=y$ corresponds to evaluating $\mathbb E[f|Y=y] = \mathbb E[f|Y]{Y=y}$, which is not meaningful because of $\mathbb E[f|Y]$ not being well-defined on such singular sets. This problem is not addressed by the introduction of regular conditional distributions, either. On the other hand it can be shown that the naively computed conditional density $f_{Z|Y=y}(z)$ (which is given by the ratio of joint and marginal densities) is a version of the conditional distribution, i.e. $\mathbb E[\{Z\in B\}|Y=y] = \int_B f_{Z|Y=y}(z) dz$ and this density can indeed be evaluated pointwise in $y$. This mismatch between mathematical theory (which generates an object which cannot produce what we need from it) and practical computation via the conditional density is an unfortunate fact. Furthermore, the classical approach does not allow a pointwise definition of conditional expectations of the form $\mathbb E[f|Y=y]$, only of conditional distributions $\mathbb E[\{Z\in B\}|Y=y]$. We propose a (as far as the author is aware) little known approach to obtaining a pointwise defined version of conditional expectation by use of the Lebesgue-Besicovich lemma without the need of additional topological arguments which are necessary in the usual derivation.
