# Solved – Tail bounds for Beta distribution

Say \$Xsimmathrm{Beta}(alpha,beta)\$. Are there any "nice" closed form upper bounds for the tail probability \$P(Xgeqepsilon)\$, that are reasonably tight when \$beta\$ is large? By "nice" I mean involving only elementary functions, and not for instance the incomplete beta function.

In my specific setting I have \$alpha=1/2\$ and \$beta=d/2\$ for some (typically large) integer \$d\$, and so one way of bounding the tail is to write \$X=frac{Y}{Y+Z}\$ where \$Ysimchi^2_1\$ and \$Zsimchi^2_d\$, and using an upper bound on \$Y\$ and lower bound on \$Z\$ (e.g. from here: What are the sharpest known tail bounds for \$chi_k^2\$ distributed variables?). This gives an answer, but it's somewhat messy (requiring the small \$d\$ case to be handled separately, etc). Is there a cleaner way?

Thanks.

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We write \$I_{epsilon}(A,B,epsilon)\$ the incomplete beta function. Thus \$P(X>epsilon) = 1-I_{epsilon}(A,B,epsilon)\$

A simple bound is

\$\$frac{2sqrt{epsilon}}{2sqrt{epsilon}+Bleft(frac{1}{2},frac{d}{2}right)}< I_epsilonleft(frac{1}{2},frac{d}{2},epsilonright)<frac{2 sqrt{epsilon}}{Bleft(frac{1}{2},frac{d}{2}right)}\$\$

A tighter lower bound (so upper bound for the tail) would be

\$\$frac{2 left(sqrt{epsilon} Bleft(frac{1}{2},frac{d}{2}right)+2 epsilonright)}{2 sqrt{epsilon} Bleft(frac{1}{2},frac{d}{2}right)+Bleft(frac{1}{2},frac{d}{2}right)^2+4 epsilon}\$\$

Mathematica will spit them out for you with

``1 - 1 / Normal[Series[1/(1 - BetaRegularized[epsilon, 1/2, d/2]), {epsilon, 0, n}]] ``

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