please forgive my lack of math terminology being a developer myself. I am looking for the equation of a bell-shaped (gaussian-like) curve with the following characteristics:
- The X domain is -infinity to +infinity with real (discrete) numbers
- The Y domain is 0 to 1
- The function is defined by only 2 parameters (like $\mu$ and $\sigma$ in the gaussian case, I will use this terminology even if it may not accurately apply)
- The sum of all $y$ equals 1 (over all X)
- The sum of all $y$ in [$\mu-\sigma$, $\mu+\sigma$] equals 0.5
- Both points $\mu-\sigma$ and $\mu+\sigma$ are inflection points
- When $\sigma$ tends to $0$, the value $y(\mu)$ tends to $1$ but will not reach 0.5 as per previous hypothesis. Call the $\sigma=0$ an edge case.
- (I suppose, when $\sigma$ tends to infinity, the value $y(\mu)$ tends to $0$ but again will not reach it because of the sum requirement)
Is such a curve possible at all ?
Can you provide it :) ?
Looking for Gaussian does not lead to what I am looking for, do you have directions or keywords for it ?