For the standard normal distribution function `F`, the value at `t` is the integral of the density function over all values less than `t`:
| `F text[(] t text[)]` | `=int_(- oo)^(t) ce^(-s^2) ds` |
| `= int_(- oo)^0 ce^(-s^2/2) ds + int_0^t ce^(-s^2/2) ds`. |
But `int_(- oo)^0 ce^(-s^2/2) ds = F text[(] 0 text[)] = 1/2,`so `F text[(] t text[)] = 1/2 + int_0^t ce^(-s^2/2) ds`.