= Differential entropy of a squared uniform variable
{title2=$h_e(X^2)=\ln2-1,\quad X\sim U(0,1)$}
Take $X$ with <continuous uniform distribution> on $(0,1)$ and set $Y=X^2$. This is an increasing invertible transformation on that interval. Its <cumulative distribution function> is $F_Y(y)=\sqrt y$, giving
$$
f_Y(y)=\frac1{2\sqrt y},\qquad 0<y<1.
$$
Although the <probability density> diverges near zero, it is <integrable>. The logarithmic stretch is integrable too:
$$
h_e(Y)=h_e(X)+\int_0^1\ln(2x)\,dx
=\boxed{\ln2-1\approx-0.306853\text{ nats}.}
$$
The <inverse function> $X=\sqrt Y$ recovers every input in the interval. The changed <differential entropy> records a changed coordinate volume, rather than a failure of recovery.
// Target: probability-and-statistics.bigb
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