I need to know if is it possible to find the mean of a normal distribution knowing the ratio of the area under the "bell" curve on the left and on the right of a given point.
To be more precise I need to solve the following equation
where I know the ratio , the variance and the point is fixed. i need to find the mean of the distribution that achieve the ratio of between the two portion of area.
thanks in advance.
actually...solving the integral equation...
with the new variable
the former become
(note that in the previous the constant terms that appear on both sides are omitted)
solving the integrals become
at the end, it was quite straightforward.
i've tested it numerically...works.
exact solution I had, however I was trying to find an analytical expression for inv_erf(x), do you think it exists? I'm gonna keep working at it!
great problem through, I'm consider a Laplace transform technique... hopefully it will point me in the right direction
i.e. you have
erf( (x-mu/(sqrt(2)*sigma) ) = (K- 1)/(K + 1)
this is obviously an integral equation which when utilising Laplace (or Fourier transforms) may yield an analytical solution.
Cheers again for the problem, really enjoyed working on it!
ps - I do have one discrepency with you solution,
I have erf(...) = (1-K)/(1+K)
on your second line, are you sure you have the K on the correct side?
to the best of my knowledge, there is no closed form for the inverse error function. i've look in some texts but the only representation was the same you can find in wikipedia
Error function - Wikipedia, the free encyclopedia
...a series expansion...
at the moment, for my work it's fine, but if you go ahead with the work...good luck1
I hope you dont take offence, Ive just played with the algebra a few times and always end up with 1 - K.
Keep posts like this coming!