Chi-Square Goodness of Fit

Is a die fair? Are signups spread evenly across weekdays? The goodness-of-fit statistic compares observed counts $O$ with expected counts $E$ :

\chi^2 = \sum \frac{(O - E)^2}{E}

Implement gof(observed, expected_probs) returning (chi2, dof, p_value), where expected counts are n * p and dof = k - 1. For the p-value, use the regularised upper incomplete gamma function $Q(\text{dof}/2,\ \chi^2/2)$ , computed with this series for $P(a, x)$ (then $Q = 1 - P$ ):

P(a, x) = \frac{x^a e^{-x}}{\Gamma(a+1)}\sum_{n=0}^{\infty}\frac{x^n}{(a+1)(a+2)\cdots(a+n)}

Sum until terms fall below 1e-15 relative to the total.