Trying to find the equation for this graph

Question

I am trying to find an equation to solve the graph below (about). Anything close to this will do as i can refine it, but i need someone to put me on the right direction.

Ideally $x$ and $f(x)$ will be $0$ to $1$. The graph would look like this

Any help would be great.

Many thanks

Answer 1

Have a look on the plot of $$f(x) = \frac{1}{\sqrt{2\pi \sigma^{2}}}e^{-\frac{x^2}{2\sigma^2}}$$

where you can try several values for $\sigma^2 > 0$.

Answer 2

I did my best trying to fit all the points using the simplest function

Hope it is useful

Answer 3

Consider the model to be $$y = \frac{1}{\sqrt{2\pi \sigma^{2}}}e^{-\frac{x^2}{2\sigma^2}}$$ Take logarithms $$\log(y)=\alpha+\beta \,x^2$$ So perform the linear regression. Back to the parameters $$\alpha=-\log({\sqrt{2\pi \sigma^{2}}})\implies \sigma^{2}=\frac{e^{-2 \alpha }}{2 \pi }$$ $$\beta=-\frac 1{2\sigma^2}\implies \sigma^{2}=-\frac 1{2\beta}$$ Use the geometric average of these two estimates and perform a nonlinear regression.

Answer 4

Try $$f(x)=(1-x^m)^n$$ for $m,n\ge2$. Varying $m$ and $n$, you can get various shapes, here is the one with $m=2, n=6$: