Distribution of limited Gaussian

Question

I was thinking about this problem. Let $X$ be Gaussian random variable with zero mean and unit variance, that is $X \sim \mathcal{N}(0,1)$ and let $S$ be any subset of $\mathbb{R}$. I want to know whether we can calculate $\mathbb{E}[X | X \not \in S]$? To this end, we need to know what the distribution of $X$ is given that we know $X \not \in S$. Any idea?

Answer 1

From the basic definitions, $$\mathbb{E}[X \mid X \notin S] = \frac{\mathbb{E}[X \mathbf{1}_{X \notin S}]}{\mathbb{P}(X \notin S)} = \frac{\int_{S^c} x f(x) \mathop{dx}}{\int_{S^c} f(x) \mathop{dx}},$$ where $f$ is the density of $X$. (Note that this holds regardless of the distribution of $X$.)

In the case of the standard Gaussian distribution, I don't think much can be said in general, although maybe for special cases of $S$ you can compute it.

Regarding your last sentence, you can think of the distribution of $X$ given $X \notin S$ as having density $$\tilde{f}(x) = \begin{cases}\frac{1}{\int_{S^c} f(z) \mathop{dz}}f(x) & {x \in S^c} \\ 0 & {x \in S} \end{cases}$$