Category:Definitions/Hat-Check Distribution

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This category contains definitions related to the hat-check distribution.
Related results can be found in Category:Hat-Check Distribution.


Let $X$ be a discrete random variable on a probability space $\struct {\Omega, \Sigma, \Pr}$.

Let $\Img X = \set {0, 2, 3, \ldots, n}$

Let $X$ represent the number of elements in a a totally ordered set with $n$ elements that are not in the correct order.

Then $X$ has the hat-check distribution with parameter $n$ (where $n > 0$) if and only if:

Definition 1

$X$ has the hat-check distribution with parameter $n$ if and only if:

$\ds \map \Pr {X = k} = \dfrac 1 {\paren {n - k }!} \sum_{s \mathop = 0}^k \dfrac {\paren {-1}^s} {s!}$


Definition 2

$X$ has the hat-check distribution with parameter $n$ if and only if:

$\ds \map \Pr {X = k} = \dfrac {!k} {n! } \dbinom n k$

where:

$!k$ is the subfactorial of $k$
$\dbinom n k$ is the binomial coefficient.

Pages in category "Definitions/Hat-Check Distribution"

The following 4 pages are in this category, out of 4 total.