= Arcsine quantiles and moments
{title2=$Q(p)=\sin^2(\pi p/2)$}
Inverting $F(x)=p$ gives the <quantile function>
$$
Q(p)=\sin^2\left(\frac{\pi p}{2}\right),\qquad0\leq p\leq1.
$$
Consequently, <inverse transform sampling> uses $X=\sin^2(\pi U/2)$ with $U$ uniform on $(0,1)$. The <quartiles> are $(2-\sqrt2)/4$, $1/2$ and $(2+\sqrt2)/4$.<Random: Probability, Mathematical Statistics, Stochastic Processes by Kyle Siegrist/The Arcsine Distribution>[Random: Probability, Mathematical Statistics, Stochastic Processes by Kyle Siegrist; The Standard Arcsine Distribution, Distribution Functions and Related Distributions: quantiles, quartiles and random quantile construction]{ref}
The first and third <quartiles> are about $0.1464$ and $0.8536$, so the central half of the <probability> spans about $0.7071$ of the unit interval. Under a <uniform distribution>, its width would be $0.5$.
The <symmetric probability distribution> gives $\mathbb E[X]=1/2$. The angle representation gives a short calculation of the second <moment>:
$$
\mathbb E[X^2]=\frac2\pi\int_0^{\pi/2}\sin^4\theta\,d\theta=\frac38,
\qquad \operatorname{Var}(X)=\frac38-\frac14=\frac18.
$$
Use $\sin^4\theta=(3-4\cos2\theta+\cos4\theta)/8$ to evaluate the integral. A <uniform distribution> on $[0,1]$ has <variance> $1/12$; the larger <variance> $1/8$ quantifies the greater spread toward the ends.
More generally, for a nonnegative <integer> $n$, the <beta function> yields
$$
\mathbb E[X^n]=\frac{B(n+1/2,1/2)}{B(1/2,1/2)}
=\frac{\binom{2n}{n}}{4^n}.
$$
For $n=0$ this is one; for $n=1,2$ it reproduces the values above. The <central binomial coefficient> thus appears in an exact moment calculation for a continuous <probability distribution>.
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