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 $P(X>x)$, $P(X<x)$ 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:$$operatorname{B}(1,,b)=frac1b,,frac{operatorname{B}(c+1,,b)}{operatorname{B}(c,,b)}=frac{c}{b+c}.$$

Answered by J.G. on December 16, 2020

0 Asked on November 6, 2021

1 Asked on November 6, 2021 by satana

martingales probability probability theory stochastic integrals stochastic processes

0 Asked on November 6, 2021 by rosco

cardinals convergence divergence functional analysis real analysis

0 Asked on November 6, 2021 by stackman

3 Asked on November 2, 2021 by s1mple

binomial coefficients combinatorics elementary number theory

1 Asked on November 2, 2021 by user806566

1 Asked on November 2, 2021 by pietro-morri

2 Asked on November 2, 2021

1 Asked on November 2, 2021

1 Asked on November 2, 2021

2 Asked on November 2, 2021 by g-g

combinatorics discrete optimization integer programming linear algebra optimization

1 Asked on November 2, 2021

1 Asked on November 2, 2021 by user731634

1 Asked on November 2, 2021 by forumwhiner

calculus characteristic functions derivatives dirac delta random variables

1 Asked on November 2, 2021

0 Asked on November 2, 2021

0 Asked on November 2, 2021

1 Asked on November 2, 2021

2 Asked on November 2, 2021 by tong_nor

0 Asked on November 2, 2021 by geromty

Get help from others!

Recent Questions

Recent Answers

- Jon Church on Why fry rice before boiling?
- Lex on Does Google Analytics track 404 page responses as valid page views?
- Joshua Engel on Why fry rice before boiling?
- haakon.io on Why fry rice before boiling?
- Peter Machado on Why fry rice before boiling?

© 2022 AnswerBun.com. All rights reserved. Sites we Love: PCI Database, MenuIva, UKBizDB, Menu Kuliner, Sharing RPP