User fido - MathOverflow

Transforming a Continuous Function

2012-09-13T17:10:59Z

Hello Everyone,

My math is quite limited so please bear with me. I will get to the point: Is there a way to transform a continuous function into a discrete one?

In essence I have a normalized Gaussian distribution defined by:

$y = b + \frac{a}{c\sqrt{\frac{\pi}{2}}} \cdot e^{\frac{-2(x-d)^{2}}{c^{2}}}$

I am only interested in values of $y$ for $0 \leq x \leq 500$. a,b,c and d are parameters.

I would like to maintain the ability of the curve to be a normalised probability distribution but only for $0 \leq x \leq 500$.

Is there a way of doing this or am I trying the impossible?

Thanks to anyone that can render assistance.

Manipulating a Gaussian Distribution

2012-09-12T17:06:14Z

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!

Comment by Fido

2012-09-13T09:14:22Z

Will this technique still work if there are other coeficcients to $e$. So if my equation is: $y = a \cdot(10 + \frac{A}{x\sqrt{\frac{pi}{2}}}) \cdot e^{-2(x-x_0)^2}/w^2$ Can I calculate the same thing for a and achieve the same effect?