$$\dfrac{\sqrt{2}\sqrt{{\pi}}\sqrt{\left|{\sigma}\right|}\sqrt[4]{w}\left(x-{\mu}\right)\mathrm{e}^\frac{\left(x-{\mu}\right)^2}{2{\sigma}^2}\left(\operatorname{erf}\left(\frac{x+\frac{v}{2}-{\mu}}{2\sqrt{\left|{\sigma}\right|}\sqrt[4]{w}}\right)-\operatorname{erf}\left(\frac{x-\frac{v}{2}-{\mu}}{2\sqrt{\left|{\sigma}\right|}\sqrt[4]{w}}\right)\right)+\sqrt{2}{\sigma}^2\mathrm{e}^\frac{\left(x-{\mu}\right)^2}{2{\sigma}^2}\left(\mathrm{e}^{-\frac{\left(x+\frac{v}{2}-{\mu}\right)^2}{4\left|{\sigma}\right|\sqrt{w}}}-\mathrm{e}^{-\frac{\left(x-\frac{v}{2}-{\mu}\right)^2}{4\left|{\sigma}\right|\sqrt{w}}}\right)}{{\sigma}\sqrt{\left|{\sigma}\right|}\left(\operatorname{erf}\left(\frac{\frac{v}{2}-{\mu}+b}{2\sqrt{\left|{\sigma}\right|}\sqrt[4]{w}}\right)-\operatorname{erf}\left(\frac{\frac{v}{2}-{\mu}+{\alpha}}{2\sqrt{\left|{\sigma}\right|}\sqrt[4]{w}}\right)\right)\sqrt[4]{w}}=0$$ Is the equation and I want w expressed as a function of v, μ, σ, α and b. Special or Elementary functions are not a problem if the result(a complicated function and not a single number) has 0 partial derivative with respect to w.
This could help me define a probability mass function with finite support and probability mass at each point of its support proportional to the density at the same value of a normally distributed random variable.
I multiplied the variance of the both discrete and finite equivalent with w divided the Probability of the respective point with the Density in the same point and set the derivative to 0.
$$\dfrac{\operatorname{erf}\left(\frac{x-({\mu}-\frac{ν}{2})}{2\sqrt[4]{{\sigma^2w}}}\right)-\operatorname{erf}\left(\frac{x-({\mu}+\frac{ν}{2})}{2\sqrt[4]{{\sigma^2w}}}\right)}{\operatorname{erf}\left(\frac{b-{\mu}+\frac{ν}{2}}{2\sqrt[4]{{\sigma^2w}}}\right)-\operatorname{erf}\left(\frac{{α - \mu}+\frac{ν}{2}}{2\sqrt[4]{{\sigma^2w}}}\right)}$$
Is the probability mass for each value of the random variable X that belongs to its support.