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In my work I met the p.d.f. of form $$p(x) \propto \frac{e^{-b \cdot x} \cdot x^c}{1 + e^{-k \cdot (x+a)}}$$ with support $x \in [0, +\infty)$ and positive parameters $b > 0, k > 0, c > 0$. It is known that integral $\int_0^{+\infty} p(x) < \infty$. Hope, $p(x)$ is for transformation of some classical distribution, so I can easily sample from $p(x)$.

If visualized, $p(x)$ is very similar to belong to the Gamma distribution family. E.g., for parameter values $a=1.0$, $b=1.7$, $c=3.0$, and $k=2.3$ function $p(x)$ is graphed below:

p.d.f. for a=1.0, b=1.7, c=3.0, and k=2.3

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  • $\begingroup$ What are the possible values of $a$? Could it be negative? Is your principal concern to sample from this distribution? $\endgroup$ – whuber Mar 21 at 20:37
  • $\begingroup$ Thanks for your response! $a$ is real, and so it may be negative. You're right, my target is to sample from $p(x)$. $\endgroup$ – Piotr Semenov Mar 21 at 20:41
  • $\begingroup$ I wondered about the sign of $a$ because when it's not too negative, rejection sampling from the Gamma distribution looks like an attractive option. But with a bit of algebraic work, it looks like rejection sampling might be effective for a wide range of the parameters--maybe even all of them. $\endgroup$ – whuber Mar 21 at 21:11

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