Expectation of uniform random variable conditioned on an event

Question

Let $D \in \mathbb R^n$ and let $X \sim \mathrm{Unif}(D)$ be a uniform random variable on $D$. Moreover, let $K \subset D$. I am struggling to write a closed form expression for the quantity $$ \mathbb E[g(X)\mid X \in K] $$ where $g\colon \mathbb R^n \to \mathbb R$ is a sufficiently smooth function. I guess I have to go through the formula $$ f_{X\mid Y}(x\mid y) = \frac{f_{X,Y}(x,y)}{f_Y(y)}, $$ with $Y = \chi_K(X)$, but I cannot conclude from here.

Answer 1

Pose $Z= \mathbb{E}[g(X)|\chi_{X \in K}]$. If we have a guess for $Z$, we just need to show that $\mathbb{E}[Z] = \mathbb{E}[g(X)]$ and that $\mathbb{E}[Z\chi_{X \in K}] = \mathbb{E}[g(X)\chi_{X \in K}]$ (of course we also need the measurability of $Z$ wrt the right $\sigma$-algebra).

I claim that $Z = \frac{\mathbb{E}[g(X)\chi_{X \in K}]}{\mathbb{P}(X \in K))} \chi_{X \in K} + \frac{\mathbb{E}[g(X)\chi_{X \in K^c}]}{\mathbb{P}(X \in K^c))} \chi_{X \in K^c} $ works (easy to verify).

So $$\mathbb{E}[g(X)|X \in K] = \frac{\mathbb{E}[g(X)\chi_{X \in K}]}{\mathbb{P}(X \in K))}.$$ Id est $X$ conditioned on $X \in K$ has law $U(K)$.