If $X^2$ follows the exponential distribution, how to prove that $X$ obeys what kind of distribution?

Question

If $X^2$ follows the exponential distribution, how to prove that $X$ obeys what kind of distribution? As we all know, if $X$ follows the Rayleigh, then $X^2$ follows the exponential distribution. Otherwise, if $X^2$ follows the exponential distribution, will $X$ only obey Rayleigh distribution?

Answer 1

You need $Y=\sqrt{X}, X \sim Exp(\lambda)$, then derive CDF of $Y$: $$ P(Y<y) = P(\sqrt{X}<y) = P(X<y^2) = P(0<X<y^2) $$ Can you handle from here?

Answer 2

If $X^2$ follows an exponential distribution (a chi-squared distribution with $2$ degrees of freedom and a suitable scale factor), then you can conclude that $|X|$ follows the Rayleigh distribution (i.e. the chi distribution with $2$ degrees of freedom and a suitable scale factor). You can use cumulative distribution functions to prove this.

You need to believe $X$ is non-negative to conclude this also is the distribution for $X$; if not then there are many possibilities.