Random spreading control by math functions

Question

I want to take control of random spreading by math functions. Here is my try with formula: $|R^{|P|}| \cdot \text{Falloff}\cdot \operatorname{sgn}(R) + \text{Offset}$

1. $R$ is random number between $-1..+1$
2. $P$ gives me control of contrast
3. Falloff is $0..1$ number that clamps random number between some range
4. Offset is shift of red line between $0..1$

But it's still not so flexible, I want also to expand inner line to range with falloff to dark side.

Such as this gradient:

Where white part of gradient is maximal part of distribution of randomness. Gray part is medium and at black part there are no any random numbers.

Can I done this with math formulas? If so, want to see these formulas.