st.statistics - Statistical test comparing two relative frequencies

Assume they are four independent beta random variables $X_i$, with means $mu_i$.

Note that the density functions would depend on the observed samples.

We could then test the hypothesis that $mu_1mu_2>mu_3mu_4$ by setting up a quadruple integral of the joint density function over the set

${ (x_1,x_2,x_3,x_4)mid x_1x_2>x_3x_4 }$.

If this integral is small, we reject the hypothesis $mu_1mu_2>mu_3mu_4$.