# Square root of a Beta(1,1) random variable

If $$X^2 \sim \text{Beta}(1,1)$$, is there a closed form for the distribution of $$X$$? If yes, what does it look like?

And if this is not too much to ask, is there a general way to find the distribution of $$X$$ for other related situations, such as $$X^2 \sim \text{Beta}(0.5, 0.5)$$ or $$X^3 \sim \text{Beta}(1,1)$$?

If $$X^{2}\sim\operatorname{Beta}(1,1)$$ (which is a uniform distribution), then $$X^p\sim\operatorname{Kumaraswamy}(1/p, 1)$$ (see the Wikipedia page). The PDF of the resulting Kumaraswamy distribution is given by $$f(x)=\frac{x^{\frac{1}{p}-1}}{p}\qquad 0 So for the example with $$p=1/2$$ we have $$(X^2)^{1/2}=X\sim\operatorname{Kumaraswamy}(2, 1)$$ with PDF $$f(x) = 2x$$ for $$0.

Generally, if $$X\sim\operatorname{Beta}(\alpha,\beta)$$, then $$X^p$$ with $$p>0$$ has PDF $$f(x)=\frac{x^{\alpha/p-1}\left(1 - x^{1/p}\right)^{\beta- 1}}{pB(\alpha, \beta)}\qquad 0 where $$B(\alpha, \beta)$$ is the beta function.

For example, if $$X^{2}\sim\operatorname{Beta}(0.5, 0.5)$$, then $$X$$ has PDF (setting $$p=1/2, \alpha = \beta=1/2$$) $$f(x)=\frac{2}{\pi\sqrt{1 - x^{2}}}\qquad 0

You ask for a general method. Here is one.

When $$X^p$$ has a Beta$$(\alpha,\beta)$$ distribution for $$p\gt 0,$$ this means for all $$0\lt y \lt 1$$ that

$$F_X(y^{1/p}) = \Pr(X \le y^{1/p}) = \Pr(X^p \le y) = \frac{1}{B(\alpha,\beta)}\int_0^y t^{\alpha-1}(1-t)^{\beta-1}\,\mathrm{d}t.$$

Differentiating with respect to $$y$$ via the Chain Rule (at the left) and Fundamental Theorem of Calculus (at the right) reveals that

$$f_X(y^{1/p}) \frac{y^{1/p-1}}{p} = \frac{\mathrm{d}}{\mathrm{d}y} F_X(y^{1/p}) = \frac{1}{B(\alpha,\beta)}y^{\alpha-1}(1-y)^{\beta-1}$$

Solve this for $$f_X,$$ writing $$x=y^{1/p}$$ (which also lies in the interval $$(0,1)$$) to obtain

\begin{aligned} f_X(x) &= \frac{1}{B(\alpha,\beta)}y^{\alpha-1}(1-y)^{\beta-1}\,p\, y^{1-1/p}\\ &= \frac{1}{B(\alpha,\beta)}\left(x^p\right)^{\alpha-1}(1-\left(x^p\right))^{\beta-1}\,p\, \left(x^p\right)^{1-1/p}\\ &=\frac{p}{B(\alpha,\beta)}\,x^{p\alpha-1}(1-x^p)^{\beta-1}. \end{aligned}

That gives you closed forms for the density $$f_X$$ and, via integration, the distribution $$F_X.$$

Here are histograms created by drawing a sample of a million values from each of four Beta distributions and taking their $$p^\text{th}$$ roots for various $$p.$$ Over them are plotted in red the graphs of $$f_X$$ to demonstrate its correctness.

Here is the R code to use if you would like to view more examples.

#
# Describe a collection of distributions.
#
Theta <- rbind(c(alpha=3, beta=7, p=5),
c(1/2, 3/2, 5),
c(3/2, 1/2, 1/5),
c(1, 1, 1/2))
n <- 1e6 # Sample size
#
# Sample and plot each distribution.
#
par(mfrow=c(2,2))
apply(Theta, 1, function(theta) {
alpha <- theta[1]; beta <- theta[2]; p <- theta[3]
Y <- rbeta(n, alpha, beta)
X <- Y^(1/p)

hist(X, breaks=80, freq=FALSE, col="Gray")
curve(p / beta(alpha,beta) * x^(p*alpha-1) * (1-x^p)^(beta-1),