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I have encountered a probability distribution having a density $f$ with respect to the Lebesgue measure on $\mathbb{R}$ equal to $$ \forall t>0, f(t)=\frac{1-\exp(-t)-t\exp(-t)}{t^2}.$$ Does this probability belongs to some known family of distributions?

Edit: There was a mistake in the expression of the function.

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  • $\begingroup$ Since $f(10) < 0$, the function is not a density function. $\endgroup$ – 5xum Oct 4 '17 at 8:51
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The function $f$ is not a density. A function is a density if it satisfies two conditions:

  • $\forall t\in\mathbb R: f(t) \geq 0 $
  • $\int_\mathbb R f(t)dt = 1$

Neither of these conditions is satisfied by your function, since $f(t)<0$ for $t>1$ and the integral of $f$ over $\mathbb R$ does not exist:

  • near $0$, the function is bounded by $\frac{1}{t^2}$ for which the integral does not converge
  • At $\infty$, the function is bounded by $\frac{1-t}{t^2}$ for which the integral does not converge.
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  • $\begingroup$ Thank you. There was a mistake in the expression of the function. $\endgroup$ – Vincent Oct 4 '17 at 9:28

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