Is there a name for the trivial probability distribution given by $P(X=x) = 1$ for a unique $x$ and $P(X=y) = 0$ for all $y \ne x$? I know it is very trivial, but since it is the distribution that minimizes entropy, I am curious if it has a specific name. (Similar to how a group with one element is referred to as the "trivial group".)
The distribution is called the Dirac measure at $x$, often denoted by $\delta_x$. Thus, for every $A\subseteq\mathbb R$, $\delta_x(A)=1$ is $x\in A$ and $\delta_x(A)=0$ otherwise.
This distribution has no PDF and its CDF is a Heaviside function, namely, $P(X\leqslant y)=0$ if $y\lt x$ and $P(X\leqslant y)=1$ if $y\geqslant x$.