I used to think that the following held true:
If $Y\sim \chi^2$, then $\frac{1}{Y}\sim \operatorname{inv}\chi^2$, the inverse $\chi^2$-distribution. Let $\chi^2_\alpha$ denote the $\alpha$-quantile of $Y$, then $\frac{1}{\chi^2_\alpha}$ is the $\alpha$-quantile of $\frac{1}{Y}$, meaning $P\left(Y\leq \chi^2_\alpha\right)=P\left(\frac{1}{Y}\leq\frac{1}{\chi^2_\alpha}\right)=\alpha.$ Now I figured out that this cannot be true, right? But what is the relationship between the two quantiles then? (Or is it true after all?)