Excess Kurtosis of Pareto Distribution/Lemma 3
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Lemma for Excess Kurtosis of Pareto Distribution
- $6 a^2 \paren {a - 1}^2 \paren {a - 2} \paren {a - 3} \paren {a - 4} = 6 a^7 - 66 a^6 + 270 a^5 - 510 a^4 + 444 a^3 - 144 a^2$
Proof
\(\ds 6 a^2 \paren {a - 1}^2 \paren {a - 2} \paren {a - 3} \paren {a - 4}\) | \(=\) | \(\ds 6 a^2 \paren {a^2 - 2 a + 1} \paren {a^2 - 5 a + 6} \paren {a - 4}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {6 a^4 - 12 a^3 + 6 a^2} \paren {a^3 - 9 a^2 + 26 a - 24}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 6 a^7 + \paren {-54 - 12} a^6 + \paren {156 + 108 +6} a^5 + \paren {-144 - 312 - 54} a^4 + \paren {288 + 156} a^3 - 144 a^2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 6 a^7 - 66 a^6 + 270 a^5 - 510 a^4 + 444 a^3 - 144 a^2\) |
$\blacksquare$