Expectation of a Normal distribution with dependent conditions

Question

$X$ and $Z$ are two independent Normal random variables and $k$ is a constant. I am interested in the following conditional expected value $$E(X\mid X<k,X+Z>k).$$

Answer 1

Assuming standard normal. $E=\frac{1}{2\pi}\int_{-\infty}^k xe^{-\frac{x^2}{2}}(\int_{(k-x)}^\infty e^{-\frac{z^2}{2}}dz)dx$

Correction: should divide by $P(X\lt k,X+Z\gt k)=\frac{1}{2\pi}\int_{-\infty}^k e^{-\frac{x^2}{2}}(\int_{(k-x)}^\infty e^{-\frac{z^2}{2}}dz)dx$

Answer 2

The general approach is:

$$ E[X| X<k, X+Z>k] = \frac{E[X1_{X<k, X+Z>k}]}{P(X<k, X+Z>k)} $$

$$ E[X1_{X<k, X+Z>k}] = \iint_D x f_{X,Z} (x,z) dx dz$$

$$ P(X<k, X+Z>k) = \iint_D f_{X,Z} (x,z) dx dz $$

where $D =\{(x,z)\in \mathbf{R}^2 \mid x<k, z >k-x \} $.