# Variance of marginals of truncated bivariate normal distribution

I have a truncated bivariate normal distribution

$$f(x,y)=\begin{cases}\frac{1}{2\pi \sigma_{1}\sigma_{2}\sqrt{1-\rho^{2}}}\exp\left(-\frac{z}{2(1-\rho^{2})}\right) &, |x| \leq a\\0 &, |x| > a\end{cases}$$

, where $$z=\frac{x^2}{\sigma_{1}^2}+\frac{y^2}{\sigma_{2}^2}-\frac{2\rho xy}{\sigma_{1}\sigma_{2}}$$

moreover, in case $$f(x,y)$$ would be not truncated, it would be normalized.

Is there a simple way to calculate variance of marginal distributions ?

I know that if the distribution is not truncated, variance of marginal distributions are $$\sigma_{1}^2$$ and $$\sigma_{2}^2$$, but what if the distribution is truncated ?