distribution of continuous symmetric X for which $E\left(X^{2k}\right)=b^k.k!$

Question

I have a continuous symmetric random variable $X$ for which $E\left(X^{2k}\right)=b^k.k!$. Since it's symmetric, all of its odd order moments are zero. And in my case, $b=2/n$ where $n$ is a large integer (if it's needed). Does $X$ have a well-known distribution? What's the distribution of $X$?

Answer 1

@Snoop put you on the right path. Consider

$$Y=\text{sgn}(X)\sqrt{|X|}$$

where $X\sim \text{Laplace}(0,b)$.

I am too lazy to write down the steps so I'll use Mathematica:

dist = TransformedDistribution[Sign[x] Sqrt[Abs[x]], x \[Distributed] LaplaceDistribution[0, b]]

FullSimplify[Moment[dist, 2 k], k ∈ PositiveIntegers]
(* b^k k! *)

FullSimplify[Moment[dist, 2 k - 1], k ∈ PositiveIntegers]
(* 0 *)