I need analytical formula of two funcitons. On picture, I painted plot, how functions should look like.
The left tail of both function does not matter, ie behavior of functions on $(0, -\inf)$ does not matter. I need both functions to be asymptotic to 0.5, one decreaxing from [0,1], second increacing from [0,0]. I prefer one parameter of each function to control the rate of decrase.
I played with transformations of $1/x$, but with no luck. My unrealistic idea of red curve is something like $exp(-x + log(-0.5)) + 0.5$ but obviously it is not solution.
Thx for any advices. Also any suggestions and hints are very welcomed.
For the red function I would use exponential decay $ f(x) = \frac{1 + e^{-\lambda x}}{2}$ while for the blue function I would use the red function reflected along $y=0.5$ so $ g(x) = \frac{1 - e^{-\lambda x}}{2}$ in both cases $\lambda$ is the value you adjust to get different decay rates.