For a normal distribution $N(\mu,\sigma^2)$, we know its cumulative distribution function is $F(x)=\Phi(\frac{x-\mu}{\sigma})$ where $\Phi(x)$ is $cdf$ for standard normal distribution which means

$$ \Phi(x) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^x e^{\frac{-t^2}{2}}dt$$

Now suppose I know that

$$F(x_1)=y_1$$ $$F(x_2)=y_2$$ ... $$F(x_k)=y_k$$

where $(x_1,y_1),(x_2,y_2),\dots (x_k,y_k)$ is known and $x_1\le x_2 ...\le x_k$,$0\le y_1\le y_2...\le 1$ How can I use these data to fit $F(x)$, i.e to get $\mu$ and $\sigma$?

If it's hard to give a closed form, any numerical form is also welcomed.

Thank you for any suggestions!


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