I understand that the pearson correlation coefficient r can be tested using student-t distribution. Also I can understand the exact distribution of r from https://en.wikipedia.org/wiki/Pearson_correlation_coefficient
But scipy claims that it also follows the beta distribution with both alpha and beta equal to (n-2)/2 https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.pearsonr.html
I am failed to proof it.
if B is the beta function.
Why
B(0.5, (n-2)/2) = 2^(n-3) * B((n-2)/2,(n-2)/2)
Can anyone give me a proof or some articles that explain it.