## On the curve generated by plotting one sine against another

A circle? A line? Actually, it depends. It depends upon parameters like the frequency or the phase; when these change, really interesting things happen. What I mean by “against another” is that the first sine function will be the $x$-coordinate while the other, the $y$-coordinate; then, if you studied engineering you might say “$y$ vs. $x$” but if you studied biology you might be used to say “$x$ vs. $y$” (personal observation). We end up then with a curve, say $\gamma(t) = (x(t), y(t))$, where

$x(t) = A_1\sin(\omega_1 t + \phi_1)$

and

$y(t) = A_2\sin(\omega_2 t + \phi_2)$.

## An asymmetric pdf with infinite support

When was the last time you needed an asymmetric (skewed) probability density function (pdf) with infinite support? Traditional skewed  distributions like the gamma family suffer from a semi-infinite support, that is, $\mathrm{supp}(p) = [0, +\infty)$. The support, if you are out of the loop, is the set of values $x$ in the domain of $p$ such that $p(x) > 0$. Why is this inconvenient? well, I will give more details about the specific application later, meanwhile, let’s say that the fact that its derivative is discontinuous at $0$ is problematic; even more, I need my function to be at least in $C^3$, that is, to have at least $3$ continuous derivatives!.

This post constitutes a somewhat dirty solution being as unwilling as I am to review any literature in depth (I might be inventing the wheel again, some wheel, but who cares; this is a blog!). Here we go.