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".)

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

  • $\begingroup$ Why do you say that "this distribution has no PDF"? It seems to me that $\delta$ defined that way is the PDF. $\endgroup$ – Sahand Apr 16 '14 at 19:20
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    $\begingroup$ Recall that "Dirac's delta function" is actually not a function in the mathematical sense of the term and that Dirac measures are not absolutely continuous with respect to Lebesgue measure hence they have no PDF. $\endgroup$ – Did Apr 16 '14 at 19:23
  • $\begingroup$ 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. $\endgroup$ – Sahand Apr 16 '14 at 20:19
  • $\begingroup$ What do you think D stands for, in PDF? $\endgroup$ – Did Apr 16 '14 at 20:23
  • $\begingroup$ "Since the described δ, I believe, will be continuous given the discrete topology of a finite set" Ouch! You are confusing "continuous" in the topological sense with "continuous" as in "continuous distributions". Please do not reinvent the wheel and check at least once the definitions of the field you are interested in, in the present case, probability theory. $\endgroup$ – Did Apr 16 '14 at 20:25

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