2
$\begingroup$

Consider the expression $$I_{p}(\alpha,\beta+1) - I_{p}(\alpha+1,\beta) = c$$ where $I_p(a,b)$ is the regularized incomplete beta function.

Question: Given $\alpha,\beta,$ and $c>0$, what is $p$?


Attempt: I need to invert the CDF's, but the problem is the arguments of the CDFs are not the same. I'm not sure how to proceed.

$\endgroup$
0
$\begingroup$

Using the inductive property

$$I_p(\alpha,\beta+1)=I_p(\alpha,\beta)+\frac{p^{\alpha}(1-p)^{\beta}}{\beta\,B(\alpha,\beta)}$$

$$I_p(\alpha+1,\beta)=I_p(\alpha,\beta)-\frac{p^{\alpha}(1-p)^{\beta}}{\alpha\,B(\alpha,\beta)}$$

It is obtained:

$$p^{\alpha}\,(1-p)^{\beta}=c\,(\frac{1}{\alpha}+\frac{1}{\beta})\,B(\alpha,\beta)\:$$

From here there is no general solution in a closed form $\forall\,\alpha$,$\beta$ but last equation is very easy to solve numerically

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.