# Is there a name for the trivial probability distribution P(X=x) = 1 for a unique x?

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".)

• CDF of that distribution is Heaviside, and PDF is Dirac $\delta$ function. – Kaster Apr 16 '14 at 18:58
• @user88595: Would that not refer to $x$ and $y$, respectively, as opposed to the distribution? – Sahand Apr 16 '14 at 18:58
• @Kaster: I always thought the Dirac $\delta$ function has $\delta(0) = \infty$ and not $\delta(0) = 1$. Would that make it not a PDF? – Sahand Apr 16 '14 at 19:02
• Also, as far as I know, $X$ is called degenerate in this case. – fgp Apr 16 '14 at 19:06
• PDF doesn't have to have $PDF(0) = 1$. – Kaster Apr 16 '14 at 20:30

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 $y\geqslant x$.
• Why do you say that "this distribution has no PDF"? It seems to me that $\delta$ defined that way is the PDF. – Sahand Apr 16 '14 at 19:20
• What if $A$ is a finite set? In that case would you say there is a PDF? Since the described $\delta$, I believe, will be continuous given the discrete topology of a finite set. – Sahand Apr 16 '14 at 20:19