Hat-Check Distribution Gives Rise to Probability Mass Function
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Theorem
Let $X$ be a discrete random variable on a probability space $\struct {\Omega, \Sigma, \Pr}$.
Let $X$ have the hat-check distribution with parameter $n$ (where $n > 0$).
Then $X$ gives rise to a probability mass function.
Proof
By definition:
- $\Img X = \set {0, 1, \ldots, n}$
- $\ds \map \Pr {X = k} = \dfrac 1 {\paren {n - k }!} \sum_{s \mathop = 0}^k \dfrac {\paren {-1}^s} {s!}$
Then:
\(\ds \map \Pr \Omega\) | \(=\) | \(\ds \sum_{k \mathop = 0}^n \dfrac 1 {\paren {n - k }!} \sum_{s \mathop = 0}^k \dfrac {\paren {-1}^s} {s!}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{k \mathop = 0}^n \dfrac 1 {\paren {n - k }! } \dfrac {n! k!} {n! k!} \sum_{s \mathop = 0}^k \dfrac {\paren {-1}^s} {s!}\) | multiplying by $1$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{k \mathop = 0}^n \dbinom n k \dfrac {k!} {n!} \sum_{s \mathop = 0}^k \dfrac {\paren {-1}^s} {s!}\) | Definition of Binomial Coefficient | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac 1 {n!} \sum_{k \mathop = 0}^n \dbinom n k k! \sum_{s \mathop = 0}^k \dfrac {\paren {-1}^s} {s!}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac 1 {n!} n!\) | Sum over k of r Choose k by -1^r-k by Polynomial | |||||||||||
\(\ds \) | \(=\) | \(\ds 1\) |
So $X$ satisfies $\map \Pr \Omega = 1$, and hence the result.
$\blacksquare$