If $X \exp(-imX)$ is an exponentially distributed random variable with mean $k$, what is the pdf of the random variable $X$? Consider $m$ a constant and $i = \sqrt{-1}$.

closed as unclear what you're asking by Did, user91500, Laurent Duval, user99914, choco_addicted Jun 12 '16 at 11:02

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Are you sure this is what you want to ask? For $X\mathrm e^{-\mathrm imX}$ to be exponentially distributed, $X\mathrm e^{-\mathrm imX}$ must in particular be almost surely real, that is, $X$ must be in $B_m\subset\mathbb C$ almost surely, where $B_m=\{z\in\mathbb C\mid z\mathrm e^{-\mathrm imz}\in\mathbb R\}$. Pretty complicated set when $m\ne0$ and $m$ is real, don't you think?

  • They're curves related to Lambert functions in the complex plane. Quite complicated set indeed. – rajb245 Feb 8 '14 at 21:44

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