Differential entropy of a squared uniform variable ($h_e(X^2)=\ln 2-1,X\sim U(0,1)$)

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 $F_Y(y)=\sqrt{y}$, givingAlthough the probability density diverges near zero, it is integrable. The logarithmic stretch is integrable too: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