= Arcsine distribution on a bounded interval
{title2=$Y=a+(b-a)X,\quad a<b$}
Specify whether two parameters mean endpoints or location and width. With endpoints $a<b$, define $Y=a+(b-a)X$ for a standard <arcsine distribution> variable $X$. Then
$$
f_Y(y)=\frac1{\pi\sqrt{(y-a)(b-y)}},\quad a<y<b,
\qquad F_Y(y)=\frac2\pi\arcsin\sqrt{\frac{y-a}{b-a}}.
$$
This is a <location-scale family> with location $a$ and width $b-a$. Its <mean> is $(a+b)/2$ and its <variance> is $(b-a)^2/8$. A convention using parameters $a,w$ for location and width instead has right endpoint $a+w$, so it must not be read as having endpoints $a,w$.<Random: Probability, Mathematical Statistics, Stochastic Processes by Kyle Siegrist/The Arcsine Distribution>[Random: Probability, Mathematical Statistics, Stochastic Processes by Kyle Siegrist; The General Arcsine Distribution, Definition, Distribution Functions and Moments: location-plus-width convention and alternative endpoint parametrization]{ref}
For example, a chosen interval $[-2,6]$ gives <mean> $2$ and <variance> $8$, while its density still concentrates toward both endpoints. It is a bounded <probability distribution>; its endpoint singularities do not make its support infinite.
Back to article page
