Hello All,

I warn in advance my math skills are quite limited so speaking to me like I am an idiot would be appreciated.

I have a polynomial :

$y = a\cdot e^{\frac{-2(x - x_0)^2}{w^2}}$

Now I know that this is of the form of a Gaussian distribution, here is my issue:

I am only interested on the positive side i.e $x \geq 0$ and $y \geq 0$. I would like to be able to manipulate the positive side of this equation such that the area below the curve is maintained on that positive side.

In essence I would like to be able to sweep through all curves defined by the above equation using a single control parameter for example $w$.

Is there a way of me deriving from the above equation another equation where I can manipulate the curve using a single parameter and ensure that the area is maintained. (Please keep in mind I am only interested in the positive side of the curve)

Ideally I would like to be able to sweep from an extreme curve such as an exponential decay all the way to a straight line such as a uniform distribution.

I have been told I should look into gamma distributions but when I looked into such I failed to apply it to my work.

Thanks to all that can help in anyway!