Hello everybody,
The generic equation of an asymmetric Gaussian curve is :
y = k * exp[-{(x-x0)^2}/{2*sig^2}] * [1+erf({a*(x-x0)}/{sig*sqrt(2)})].
I need to know the coordinates ("x" and "y") of the top (assuming k>0) of such an asymmetric Gaussian curve, as a function of the asymmetry parameter "a". Does there exist an analytic function that gives this? At least, does there exist such a function for the more specific but also simpler "y = exp[-{x^2}/2] * [1+erf({a*x}/{sqrt(2)})]" (i.e. with k=1, sig=1, x0=0)? Then it would be easy to generalize for any k, sig and x0.
Thank you very much.
There should be a simple form. If you change k the y_max value will simply be scaled by that (x_max will not change). If you change x0 only x_max will change. The new x_max will be the old one plus x0. If you change sigma the width of the curve will broaden resulting in a less skewed curve. In the formulas everything related to a should be scaled accordingly. If you give the solution to the simple case I can write the solution for the more complicated.
Sure Thomas, if I had the solution for the simple I could also write it for the general case. However the problem is that I don't have the analytic formula even for the simple case. I tried to solve df/dx = 0 for this simple case ; however I got : X*(1+erf(X)) + (1/sqrt(pi))*exp(-X^2) = 0, where X = x/sqrt(2), and I don't find any way of solving it. Any idea ?
Okay, I stuffed it into MATLAB and there is no general solution for the simple version either.
OK, thanks for having searched it — well, I will have to write a paper defining this "Coffinet's special function" ^^
Good luck! Maybe try to consider it from a numerical point of view instead.
Yes it is what I actually decided to do, sampling "a" from -100 to +100 by step of 0.01 and "x" from -1 to +1 with a step of 0.0001, and then finding the maximum "y" and the corresponding "x". This should be enough accurate for what I have to do.
Here is a series for the maximum position in x valid for small |a|
how about this?
Article TO DETERMINE SKEWNESS, MEAN AND DEVIATION WITH A NEW APPROAC...
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