# How do I prove that the SBAF activation function is not a probability density function?

The SBAF activation function is as follows - Note : 0<=x<=1 $$f(x) = \frac{1}{1+ kx^a(1-x)^{1-a}}$$ Where k and a are constants. I know we have to show that integral $$\int_{-\infty}^{\infty} f(x)\, dx$$ is not equal to 1. But I'm not able to integrate this. Can someone help me out ?

• Well, the first problem is that the function gets negative values. That already rules out the possibility of it being a PDF. – Matti P. Apr 15 at 6:57
• I'm sorry , I forgot to mention 0<=x<=1. – Hari Charan Apr 15 at 7:05
• Also k and a are positive constants. – Hari Charan Apr 15 at 7:06

$$0\leq f(x) \leq 1$$ for all $$x \in (0,1)$$. So $$\int_0^{1}f(x)dx \leq 1$$ and equality can hold only of $$f(x)=1$$ for all $$x$$. But $$f(x) \neq 1$$ for any $$x \in (0,1)$$!.