User:Sintel/Expected Dirichlet coefficient for temperaments: Difference between revisions
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:<math> | :<math> | ||
\begin{aligned} | \begin{aligned} | ||
\mathrm{Vol}(d(X,y) | \mathrm{Vol}(d(X,y) \le r) | ||
&\approx \frac{1}{B\!\left(\frac{n-k}{2}, \frac{k}{2}\right)} \cdot \frac{r^{n-k}}{\frac{n-k}{2}} \\[10pt] | &\approx \frac{1}{B\!\left(\frac{n-k}{2}, \frac{k}{2}\right)} \cdot \frac{r^{n-k}}{\frac{n-k}{2}} \\[10pt] | ||
&= \frac{\Gamma\!\left(\frac{n}{2}\right)}{\Gamma\!\left(\frac{k}{2}\right)\,\Gamma\!\left(\frac{n-k+2}{2}\right)}\, r^{n-k} | &= \frac{\Gamma\!\left(\frac{n}{2}\right)}{\Gamma\!\left(\frac{k}{2}\right)\,\Gamma\!\left(\frac{n-k+2}{2}\right)}\, r^{n-k} | ||