# Inverse CDF of a Standard Normal Variable

In many applications, for example Monte Carlo methods, we require the inverse CDF of a standard normal random variable. But the CDF:

$$\Phi(x)= \int_{-\infty}^ x \frac{1}{\sqrt{2\pi}} e^{-t^2/2} dt$$

cannot be obtained in closed form. Is there a way to define its inverse, say $\Phi^{-1}(y)$, then? Any help is greatly appreciated. Thank you.

Since $\Phi(x)$ is strictly increasing, its inverse is well defined.
In general for a regular function $f$, in order to find x such that $f(x) = y$, there are plenty of numerical algorithms, such as Newton-Raphson method.