Conditional Expectations

In the investigation of sufficient statistics in Sect. 5.1, conditional probabilities \(\mathbb{P}^{B}:\mathcal{P}(\Omega)\to[0,1],\ A\mapsto\frac{\mathbb{P}(A\cap B)}{\mathbb{P}(B)}\) played an important role. The necessary prerequisite \(\mathbb{P}(B)> 0\) was uncritical in the discrete probability spaces considered. In order to now be able to investigate the question of sufficient statistics within the framework of general probability spaces, a generalization of the previously considered conditional probabilities is necessary. Starting from a probability space \((\Omega,\mathcal{S},\mathbb{P})\) we consider a numerical, ( \(\mathbb{P}\) -)integrable random variable \(X:\Omega\to\bar{\mathbb{R}}.\) The integral \(\mathbb{E}(X):=\int Xd\mathbb{P}\) is referred to as the expected value of X. The mapping \(Y:\Omega\to\bar{\mathbb{R}},\quad\omega\mapsto\mathbb{E}(X)\) is measurable for every σ-field \(\mathcal{G}\) over Ω \(\mathcal{G}\) - \(\bar{\mathcal{B}}\) -measurable. Thus, \(\{\emptyset,\Omega\}\) is the smallest of all σ-fields \(\mathcal{G}\) over Ω, for which Y \(\mathcal{G}\) - \(\bar{\mathcal{B}}\) -measurable, and it holds: \(\int\limits_{A}Yd\mathbb{P}=\int\limits_{A}Xd\mathbb{P}\quad{\text{f}}{\ddot{\rm{u}}}{\text{r}} \; \text{alle }A\in\{\emptyset,\Omega\}.\)