Theorem: Skewness of Hat-Check Distribution

Let $X$ be a discrete random variable with a Hat-Check distribution with parameter $n$. ($n \gt 2$)
Then the skewness $\gamma_1$ of $X$ is given by: 
:$\gamma_1 = -1$

Proof:
From Skewness in terms of Non-Central Moments: 
:$\gamma_1 = \dfrac {\expect {X^3} - 3 \mu \sigma^2 - \mu^3} {\sigma^3}$ 
where: 
:$\mu$ is the expectation of $X$. 
:$\sigma$ is the standard deviation of $X$.
We have, by Expectation of Hat-Check Distribution:
:$\expect X = n - 1$
By Variance of Hat-Check Distribution:
:$\var X = \sigma^2 = 1$
so:
:$\sigma = \sqrt 1 = 1$
To now calculate $\gamma_1$, we must calculate $\expect {X^3}$.
{{begin-eqn}}
{{eqn | l = \expect {X^3}
      | r = \sum_{k \mathop = 0}^n {k^3 \dfrac 1 {\paren {n - k }!} \sum_{s \mathop = 0}^k \dfrac {\paren {-1}^s} {s!} }
      | c = {{Defof|Hat-Check Distribution}}
}}
{{eqn | r = \sum_{k \mathop = 1}^n {k^3 \dfrac 1 {\paren {n - k }!} \sum_{s \mathop = 0}^k \dfrac {\paren {-1}^s} {s!} }
      | c = as the $k = 0$ term vanishes
}}
{{eqn | r = \sum_{y \mathop = n - 1}^0 \paren {n - y }^3 \dfrac 1 {y!} \sum_{s \mathop = 0}^{n - y} \dfrac {\paren {-1}^s} {s!}
      | c = Let $y = n - k$
}}
{{eqn | r = n^3 \sum_{y \mathop = 0}^{n - 1} \dfrac 1 {y!} \sum_{s \mathop = 0}^{n - y} \dfrac {\paren {-1}^s} {s!} - 3n^2 \sum_{y \mathop = 0}^{n - 1} \dfrac y {y!} \sum_{s \mathop = 0}^{n - y} \dfrac {\paren {-1}^s} {s!} + 3n \sum_{y \mathop = 0}^{n - 1} \dfrac {y^2} {y!} \sum_{s \mathop = 0}^{n - y} \dfrac {\paren {-1}^s} {s!} - \sum_{y \mathop = 0}^{n - 1} \dfrac {y^3} {y!} \sum_{s \mathop = 0}^{n - y} \dfrac {\paren {-1}^s} {s!}
      | c = 
}}
{{eqn | r = n^3 \sum_{k \mathop = 1}^n \dfrac 1 {\paren {n - k }!} \sum_{s \mathop = 0}^k \dfrac {\paren {-1}^s} {s!} - 3n^2 \sum_{k \mathop = 1}^n \dfrac {n - k} {\paren {n - k }!} \sum_{s \mathop = 0}^k \dfrac {\paren {-1}^s} {s!} + 3n \sum_{k \mathop = 1}^n \dfrac {\paren {n - k}^2 } {\paren {n - k }!} \sum_{s \mathop = 0}^k \dfrac {\paren {-1}^s} {s!} - \sum_{k \mathop = 1}^n \dfrac {\paren {n - k}^3 } {\paren {n - k }!} \sum_{s \mathop = 0}^k \dfrac {\paren {-1}^s} {s!}
      | c = Let $y = n - k$
}}
{{eqn | r = n^3 \sum_{k \mathop = 1}^n \dfrac 1 {\paren {n - k }!} \sum_{s \mathop = 0}^k \dfrac {\paren {-1}^s} {s!} - 3n^2 \sum_{k \mathop = 1}^n \dfrac {n - k} {\paren {n - k }!} \sum_{s \mathop = 0}^k \dfrac {\paren {-1}^s} {s!} + 3n \sum_{k \mathop = 1}^n \dfrac {\paren {n - k}^2 } {\paren {n - k }!} \sum_{s \mathop = 0}^k \dfrac {\paren {-1}^s} {s!} - \sum_{k \mathop = 1}^n \dfrac {\paren {n - k}^3 } {\paren {n - k }!} \sum_{s \mathop = 0}^k \dfrac {\paren {-1}^s} {s!} + 4 \dfrac {n^3} {n!} - 4 \dfrac {n^3} {n!}
      | c = adding $0$ - re-index sums
}}
{{eqn | r = n^3 \sum_{k \mathop = 0}^n \dfrac 1 {\paren {n - k }!} \sum_{s \mathop = 0}^k \dfrac {\paren {-1}^s} {s!} - 3n^2 \sum_{k \mathop = 0}^n \dfrac {n - k} {\paren {n - k }!} \sum_{s \mathop = 0}^k \dfrac {\paren {-1}^s} {s!} + 3n \sum_{k \mathop = 0}^n \dfrac {\paren {n - k}^2 } {\paren {n - k }!} \sum_{s \mathop = 0}^k \dfrac {\paren {-1}^s} {s!} - \sum_{k \mathop = 0}^n \dfrac {\paren {n - k}^3 } {\paren {n - k }!} \sum_{s \mathop = 0}^k \dfrac {\paren {-1}^s} {s!}
      | c = 
}}
{{eqn | r = n^3 \sum_{k \mathop = 0}^n \dfrac 1 {\paren {n - k }!} \sum_{s \mathop = 0}^k \dfrac {\paren {-1}^s} {s!} - 3n^2 \sum_{k \mathop = 0}^{n - 1} \dfrac {n - k} {\paren {n - k }!} \sum_{s \mathop = 0}^k \dfrac {\paren {-1}^s} {s!} + 3n \sum_{k \mathop = 0}^{n - 1} \dfrac {\paren {n - k}^2 } {\paren {n - k }!} \sum_{s \mathop = 0}^k \dfrac {\paren {-1}^s} {s!} - \sum_{k \mathop = 0}^{n - 1} \dfrac {\paren {n - k}^3 } {\paren {n - k }!} \sum_{s \mathop = 0}^k \dfrac {\paren {-1}^s} {s!}
      | c = as the $k = n$ term vanishes
}}
{{eqn | r = n^3 \sum_{k \mathop = 0}^n \dfrac 1 {\paren {n - k }!} \sum_{s \mathop = 0}^k \dfrac {\paren {-1}^s} {s!} - 3n^2 \sum_{k \mathop = 0}^{n - 1} \dfrac 1 {\paren {n - 1 - k }!} \sum_{s \mathop = 0}^k \dfrac {\paren {-1}^s} {s!} + 3n \sum_{k \mathop = 0}^{n - 1} \dfrac {\paren {n - k} } {\paren {n - 1 - k }!} \sum_{s \mathop = 0}^k \dfrac {\paren {-1}^s} {s!} - \sum_{k \mathop = 0}^{n - 1} \dfrac {\paren {n - k}^2 } {\paren {n - 1 - k }!} \sum_{s \mathop = 0}^k \dfrac {\paren {-1}^s} {s!}
      | c = canceling terms
}}
{{eqn | r = n^3 \sum_{k \mathop = 0}^n \dfrac 1 {\paren {n - k }!} \sum_{s \mathop = 0}^k \dfrac {\paren {-1}^s} {s!} - 3n^2 \sum_{k \mathop = 0}^{n - 1} \dfrac 1 {\paren {n - 1 - k }!} \sum_{s \mathop = 0}^k \dfrac {\paren {-1}^s} {s!} + 3n^2 \sum_{k \mathop = 0}^{n - 1} \dfrac 1 {\paren {n - 1 - k }!} \sum_{s \mathop = 0}^k \dfrac {\paren {-1}^s} {s!} - 3n \sum_{k \mathop = 0}^{n - 1} \dfrac k {\paren {n - 1 - k }!} \sum_{s \mathop = 0}^k \dfrac {\paren {-1}^s} {s!}
      | c = 
}}
{{eqn | o =
      | ro = -
      | r = n^2 \sum_{k \mathop = 0}^{n - 1} \dfrac 1 {\paren {n - 1 - k }!} \sum_{s \mathop = 0}^k \dfrac {\paren {-1}^s} {s!} + 2n \sum_{k \mathop = 0}^{n - 1} \dfrac k {\paren {n - 1 - k }!} \sum_{s \mathop = 0}^k \dfrac {\paren {-1}^s} {s!} - \sum_{k \mathop = 0}^{n - 1} \dfrac {k^2 } {\paren {n - 1 - k }!} \sum_{s \mathop = 0}^k \dfrac {\paren {-1}^s} {s!}
      | c = 
}}
{{end-eqn}}
To help complete the sum above, recall that:
{{begin-eqn}}
{{eqn | l = \sum_{k \mathop = 0}^n \dfrac 1 {\paren {n - k }!} \sum_{s \mathop = 0}^k \dfrac {\paren {-1}^s} {s!}
      | r = 1
      | c = Hat-Check Distribution Gives Rise to Probability Mass Function
}}
{{eqn | l = \sum_{k \mathop = 0}^{n - 1} \dfrac 1 {\paren {n - 1 - k }!} \sum_{s \mathop = 0}^k \dfrac {\paren {-1}^s} {s!}
      | r = 1
      | c = Hat-Check Distribution Gives Rise to Probability Mass Function
}}
{{eqn | l = \sum_{k \mathop = 0}^{n - 1} \dfrac k {\paren {n - 1 - k }!} \sum_{s \mathop = 0}^k \dfrac {\paren {-1}^s} {s!}
      | r = \paren {n - 1} - 1
      | c = Expectation of Hat-Check Distribution
}}
{{eqn | r = \paren {n - 2}
      | c = 
}}
{{eqn | l = \sum_{k \mathop = 0}^{n - 1} \dfrac {k^2} {\paren {n - 1 - k }!} \sum_{s \mathop = 0}^k \dfrac {\paren {-1}^s} {s!}
      | r = \paren{\paren {n - 1} - 1}^2 + 1
      | c = Variance of Hat-Check Distribution
}}
{{eqn | r = \paren{n^2 -4n + 5}
      | c = 
}}
{{end-eqn}}
Therefore:
{{begin-eqn}}
{{eqn | r = n^3 - 3n^2 + 3n^2 - 3n \paren {n - 2} - n^2 + 2n \paren {n - 2} - \paren {n^2 - 4n + 5}
      | c =  
}}
{{eqn | r = n^3 + \paren {- 3 + 3 - 3 - 1 + 2 - 1}n^2 + \paren {6 - 4 + 4}n - 5
      | c = 
}}
{{eqn | r = n^3 - 3n^2 + 6n - 5
      | c = 
}}
{{end-eqn}}
So:
{{begin-eqn}}
{{eqn	| l = \gamma_1
	| r = \frac {\paren {n^3 - 3n^2 + 6n - 5 } - 3 \paren {n - 1} 1 - \paren {n - 1}^3} {1^{3/2} }
}}
{{eqn	| r = \paren {n^3 - 3n^2 + 6n - 5 } - 3n + 3 - \paren {n^3 - 3n^2 + 3n - 1}
}}
{{eqn	| r = -1
}}
{{end-eqn}}
{{qed}}
Category:Skewness
Category:Hat-Check Distribution
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