# What distribution has the p.d.f. of form $\frac{e^{-b \cdot x} \cdot x^c}{1 + e^{-k \cdot (x+a)}}$ for $x >0$ and parameters $b > 0, k > 0, c > 0$?

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:

• What are the possible values of $a$? Could it be negative? Is your principal concern to sample from this distribution? – whuber Mar 21 at 20:37
• Thanks for your response! $a$ is real, and so it may be negative. You're right, my target is to sample from $p(x)$. – Piotr Semenov Mar 21 at 20:41
• 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. – whuber Mar 21 at 21:11