Excess Kurtosis of Pareto Distribution/Lemma 1

Lemma for Excess Kurtosis of Pareto Distribution

$\paren {a - 1}^4 \paren {a - 2}^2 \paren {a - 3} = a^7 - 11 a^6 + 50 a^5 - 122 a^4 + 173 a^3 - 143 a^2 + 64 a - 12$

Proof

\(\ds \paren {a - 1}^4 \paren {a - 2}^2 \paren {a - 3}\)

\(=\)

\(\ds \paren {a^4 - 4 a^3 + 6 a^2 - 4 a + 1} \paren {a^2 - 4 a + 4} \paren {a - 3}\)

\(\ds \)

\(=\)

\(\ds \paren {a^4 - 4 a^3 + 6 a^2 - 4 a + 1} \paren {a^3 - 4 a^2 + 4 a - 3 a^2 + 12 a - 12}\)

\(\ds \)

\(=\)

\(\ds \paren {a^4 - 4 a^3 + 6 a^2 - 4 a + 1} \paren {a^3 - 7 a^2 + 16 a - 12}\)

\(\ds \)

\(=\)

\(\ds a^7 + \paren {-7 - 4} a^6 + \paren {16 + 28 +6} a^5 + \paren {-12 - 64 - 42 - 4} a^4 + \paren {48 + 96 + 28 + 1} a^3 + \paren {-72 - 64 - 7} a^2 + \paren {48 + 16} a - 12\)

\(\ds \)

\(=\)

\(\ds a^7 - 11 a^6 + 50 a^5 - 122 a^4 + 173 a^3 - 143 a^2 + 64 a - 12\)

$\blacksquare$