Does this probability density function have a name?

Question

I have a random variable distributed according to the density function

$$P(x)=|x|e^{-x^2}$$

Does this pdf have a name?

Answer 1

The function $f(x;\sigma)=\frac{x}{\sigma^2} e^{-x^2/(2\sigma^2)}$ for $x \geq 0$ is known as the Rayleigh distribution. What you have is an adaptation of the Rayleigh distribution.

Call them modified or adapted Rayleigh distribution.

Answer 2

Formally, it is a Reflected Rayleigh distribution ... with the Rayleigh parameter parameter $\sigma = \frac{1}{\sqrt2}$

Similary, other positive random variables that are reflected about 0 to yield a variable on the real line are the Reflected Gamma, Reflected Weibull etc