How to find conditional expectation $E(X\mid Y,Z>c)$?

Question

I have three random variables (X, Y, Z) as follows:

$X \sim N(0,1)$,

$Y = X+\varepsilon_1$, where $\varepsilon_1 \sim N(0,q)$,

$Z = X+\varepsilon_2$, where $\varepsilon_2 \sim N(0,r)$.

I want to calculate expectation $E[X\mid Y, Z>c]$.

I found the following useful links: 1)How to find conditional expectation $E(X|Y,Z)$? and 2) Expected value of $X$ given $X > Y$

I understand that (X,Y,Z) are jointly normal random variables. Am I right if I calculate the following:

$$\int_{-\infty}^{\infty} \int_{c-\varepsilon_2}^{\infty} x f_{X\mid Y,Z}(X\mid Y=y, Z=z)\,dx\, dz,$$

where $f_{X\mid Y,Z}(X\mid Y=y, Z=z) = \frac{f_{X,Y,Z}(X=x,Y=y, Z=z)}{f_{Y,Z}(Y=y, Z=z)}?$

Answer 1

Note that $$\Pr\{X\le x|Y,Z>c\}={\Pr\{X\le x,Y>c,Z>c\}\over \Pr\{Y,Z>c\}}$$from which$$f_{X|Y,Z>c}(x)={\int_c^\infty \int_c^\infty f_{XYZ}(x,y,z)dydz\over \int_{-\infty}^\infty\int_c^\infty \int_c^\infty f_{XYZ}(x,y,z)dydz}$$Afterward, the final step is calculating $$\int_{-\infty}^\infty xf_{X|Y,Z>c}(x)dx$$