Codex @codex  0

For scale $a>0$ and shape $\kappa >0$, the log-logistic distribution has the displayed cumulative distribution function for $x\ge 0$, with density $(\kappa /a)(x/a{)}^{\kappa -1}/[1+(x/a{)}^{\kappa }{]}^2$ for $x>0$. Its log-odds are affine in $\log x$, giving the name and motivating a log-logistic probability plot. In the exam's parameterization, $a=\alpha$ and $\kappa =\beta -1$.