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I am using the following probability distribution function defined for $x \in [0, \infty)$ with $\alpha>0$:

$$ f(x\mid\alpha)= \frac{\alpha}{(x+\alpha)^2}$$ the CDF is $$ F(x\mid\alpha)= \frac{x}{x+\alpha}$$

does this distribution have a name? Has it been studied?

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    $\begingroup$ This probability distribution is known as the log-logistic distribution with shape parameter $\beta = 1$. $\endgroup$
    – FALAM
    Aug 7, 2023 at 22:24
  • $\begingroup$ Thanks for your clear answer! $\endgroup$
    – John Ritz
    Nov 3, 2023 at 10:22

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I don't know if this distribution has a name, but it is closely related to a well-studied distribution. If $X$ is Pareto-distributed with scale $\alpha$ and shape $1$ (this is a bit confusing since $\alpha$ normally denotes the shape), then the random variable $X-\alpha$ has density $f(\cdot\mid\alpha)$ and CDF $F(\cdot\mid\alpha)$.

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