# Is there closed formula to calculate the probability of beta distribution?

Mathematics Asked by vesii on December 16, 2020

Part of my investigation of the properties of the beta distribution, I was trying to figure if for $$Xsim Beta(a,b)$$ is there a closed formula for $$P(X>x)$$, $$P(X and $$P(X=x)$$? For example, if $$Xsimexp(lambda)$$ then we know that $$P(X >a)=e^{-lambda a}$$ and $$P(Xleq a)=1-e^{-lambda a}$$. Is there something similar for beta distribution?

In general, we need a special function for $$P(Xle x)$$. However, if $$a$$ and/or $$b$$ is a positive integer, we can obtain a closed-form solution (indeed, if both are positive integers it's a polynomial). Suppose $$a$$ is an integer:begin{align}int_0^x(1-t)^{b-1}dt&=frac{1-(1-x)^b}{b},\int_0^xt^{c+1}(1-t)^{b-1}dt&=int_0^xt^c(1-t)^{b-1}dt-int_0^xt^c(1-t)^bdt.end{align}Further, the normalization constant that ensures probabilities sum to $$1$$ will still be a value of the Beta function. Suppose $$a$$ is an integer:$$operatorname{B}(1,,b)=frac1b,,frac{operatorname{B}(c+1,,b)}{operatorname{B}(c,,b)}=frac{c}{b+c}.$$