What is the expected value of a normally distributed random variable, conditional on its squared value being higher than some number?

Question

Let X be normally distributed random variable with some given mean and variance. Is there an expression for the expected value of X, conditional on X^2 being greater than some constant c?

Answer 1

$\Pr(X^2 \gt c) = \Pr(X \gt \sqrt{c})+ \Pr(X \gt \sqrt{-c})$ so this is related to a truncated normal distribution though you are looking to truncate inside rather than outside.

So I would have thought that if $\alpha=\dfrac{-\sqrt{c}-\mu}{\sigma}$ and $\beta=\dfrac{\sqrt{c}-\mu}{\sigma}$ than the mean you are looking for will be $$E[X \mid X^2 \gt c]=\mu + \dfrac{\phi(\beta)-\phi(\alpha)}{1-\Phi(\beta)+\Phi(\alpha)}\sigma$$ where $\phi(x)$ is the density of a standard normal and $\Phi(x)$ is the cumulative distribution function of a standard normal