What is the probability distribution of this random variable? Let $X$ be a random variable $\mathcal N(0,1)$.
How can we find the distribution of $$Y= \frac{1}{|X| \sqrt{2 \pi}} e^{\frac{-1}{2 X^2}}$$
What are the available tools to solve this problem, or any similar problem, where we want to determine the distribution of $f(X)$ where $f$ is not explicitly invertible in $X$ ?
 A: When a random variable X, is transformed by a monotonic function $Y=Y(X)$ it means that 
$$P(x < X < x+dx) = P(y < Y < y+dy)$$ 
where $y = y(x)$.
In words, the probability that an outcome of $X$ falls within the range $(x,x+dx)$, equals the probability that an outcome of $Y$ falls within the corresponding range $(y,y+dy)$.
Rewriting that equation in terms of probability density functions gives,
$$
f(x)\,dx = g(y)\, dy \ .
$$
The transformation you are interested in is not monotonic, but it is monotonic for $X>0$ and is symmetric upon changed of sign of $X$.
In this case, the equation is modified by considering only $X>0$ and including a factor of 2 when calculating the density $g(y)$. The resulting relation is:
$$
g(y) = 2 f(x) \left\vert \frac{dy}{dx} \right\vert
$$
To evaluate this, you will have to take the derivative of your equation (you can do that analytically), and also evaluate $x(y)$ (probably have to do that numerically).
You can find more detailed information in texts and online, typically under the heading of "Transformation of Random Variables".
