### Math proof required (Probability)

FromPr[E] = Pr[E|A].Pr[A] + Pr[E|A'].Pr[A']how can we provePr[E] <= Pr[E|A] + Pr[A']...Read more

### probability theory - Substitute Concrete Value in Conditional Expectation

Let $(\Omega, \mathcal{G}, \mathbb{P})$ be a probability space.Let $$X, Y : \Omega \rightarrow \mathbb{R}$$ be random variables.Furthermore, let$$f: \mathbb{R}^2 \rightarrow \mathbb{R}$$be a $\mathcal{B}(\mathbb{R}^2)/\mathcal{B}(\mathbb{R})$-measurable function such that, for all $y \in \mathbb{R}$, the random variables $f(X,y)$ and $f(X,Y)$ have finite expectation.Now let $y \in \mathbb{R}$ be arbitrary. Under the above assumptions, the expected value $\mathbb{E}[f(X,y)]$ and a $\mathbb{P}$-unique conditional expectation \$\mathbb{E}[f(X...Read more