Excess Kurtosis of Pareto Distribution/Lemma 2

Lemma for Excess Kurtosis of Pareto Distribution

$4 a \paren {a - 1}^3 \paren {a - 2}^2 \paren {a - 4} = 4 a^7 - 44 a^6 + 188 a^5 - 404 a^4 + 464 a^3 - 272 a^2 + 64 a$

Proof

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

\(=\)

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

\(\ds \)

\(=\)

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

\(\ds \)

\(=\)

\(\ds \paren {4 a^4 - 12 a^3 + 12 a^2 - 4 a } \paren {a^3 - 8 a^2 + 20 a - 16}\)

\(\ds \)

\(=\)

\(\ds 4 a^7 + \paren {-32 - 12} a^6 + \paren {80 + 96 + 12} a^5 + \paren {-64 - 240 - 96 - 4} a^4 + \paren {192 + 240 + 32} a^3 + \paren {-192 - 80} a^2 + 64 a\)

\(\ds \)

\(=\)

\(\ds 4 a^7 - 44 a^6 + 188 a^5 - 404 a^4 + 464 a^3 - 272 a^2 + 64 a\)

$\blacksquare$