if $X$ ~ $Beta(a,b)$ then what is $P(X < \frac{1}{4})$?

Question

I have tried many ways but I can not understand how to calculate that probability when a and b are not specified, I never had a problem in solving probability problems until now but this one I can not understand and can not solve
I know what is Beta(a, b) and what is probability density function for beta distribution and I wrote $P(X < \frac{1}{4}) = \frac{\int_{0}^{\frac{1}{4}}x^{a-1}(1-x)^{b-1}dx}{\int_{0}^{1}x^{a-1}(1-x)^{b-1}dx}$ but I can now go farther
and another question is, is this a hard question or I'm bad at this or I'm missing something? Thanks in advance for your help

Answer 1

You are looking for the cdf of the beta distribution. There is no way to simplify your expression, and indeed it has the special name regularised incomplete beta function and short symbol $I_x(a,b)$; your probability would thus be $I_{1/4}(a,b)$.