Excess Kurtosis of Beta Distribution/Lemma 1

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Lemma for Excess Kurtosis of Beta Distribution

\(\ds \paren {\alpha + 1} \paren {\alpha + 2} \paren {\alpha + 3} \paren {\alpha + \beta}^3 \paren {\alpha + \beta + 1}\) \(=\) \(\ds \alpha^7 + \paren {4 \beta + 7} \alpha^6 + \paren {6 \beta^2 + 27 \beta + 17} \alpha^5 + \paren {4 \beta^3 + 39 \beta^2 + 62 \beta + 17} \alpha^4 + \paren {\beta^4 + 25 \beta^3 + 84 \beta^2 + 57 \beta + 6} \alpha^3\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds \paren {6 \beta^4 + 50 \beta^3 + 69 \beta^2 + 18 \beta} \alpha^2 + \paren {11 \beta^4 + 35 \beta^3 + 18 \beta^2} \alpha + 6 \beta^4 + 6 \beta^3\)


Proof

\(\ds \paren {\alpha + 1} \paren {\alpha + 2} \paren {\alpha + 3} \paren {\alpha + \beta}^3 \paren {\alpha + \beta + 1}\) \(=\) \(\ds \paren {\alpha + 1} \paren {\alpha + 2} \paren {\alpha + 3} \paren {\paren {\alpha + \beta}^4 + \paren {\alpha + \beta}^3 }\) Group $\paren {\alpha + \beta}$
\(\ds \) \(=\) \(\ds \paren {\alpha + 1} \paren {\alpha + 2} \paren {\alpha \paren {\alpha + \beta}^4 + \alpha \paren {\alpha + \beta}^3 + 3 \paren {\alpha + \beta}^4 + 3 \paren {\alpha + \beta}^3}\) Distribute $\paren {\alpha + 3 }$
\(\ds \) \(=\) \(\ds \paren {\alpha + 1} \paren {\alpha^2 \paren {\alpha + \beta}^4 + \alpha^2 \paren {\alpha + \beta}^3 + 3 \alpha \paren {\alpha + \beta}^4 + 3 \alpha \paren {\alpha + \beta}^3 + 2 \alpha \paren {\alpha + \beta}^4 + 2 \alpha \paren {\alpha + \beta}^3 + 6 \paren {\alpha + \beta}^4 + 6 \paren {\alpha + \beta}^3}\) Distribute $\paren {\alpha + 2 }$
\(\ds \) \(=\) \(\ds \paren {\alpha + 1} \paren {\alpha^2 \paren {\alpha + \beta}^4 + \alpha^2 \paren {\alpha + \beta}^3 + 5 \alpha \paren {\alpha + \beta}^4 + 5 \alpha \paren {\alpha + \beta}^3 + 6 \paren {\alpha + \beta}^4 + 6 \paren {\alpha + \beta}^3}\) $3 \alpha + 2 \alpha = 5 \alpha$
\(\ds \) \(=\) \(\ds \alpha^3 \paren {\alpha + \beta}^4 + \alpha^3 \paren {\alpha + \beta}^3 + 5 \alpha^2 \paren {\alpha + \beta}^4 + 5 \alpha^2 \paren {\alpha + \beta}^3 + 6 \alpha \paren {\alpha + \beta}^4 + 6 \alpha \paren {\alpha + \beta}^3\) Distribute $\paren {\alpha + 1 }$
\(\ds \) \(\) \(\, \ds + \, \) \(\ds \alpha^2 \paren {\alpha + \beta}^4 + \alpha^2 \paren {\alpha + \beta}^3 + 5 \alpha \paren {\alpha + \beta}^4 + 5 \alpha \paren {\alpha + \beta}^3 + 6 \paren {\alpha + \beta}^4 + 6 \paren {\alpha + \beta}^3\)
\(\ds \) \(=\) \(\ds \alpha^3 \paren {\alpha + \beta}^4 + \alpha^3 \paren {\alpha + \beta}^3 + 6 \alpha^2 \paren {\alpha + \beta}^4 + 6 \alpha^2 \paren {\alpha + \beta}^3 + 11 \alpha \paren {\alpha + \beta}^4 + 11 \alpha \paren {\alpha + \beta}^3 + 6 \paren {\alpha + \beta}^4 + 6 \paren {\alpha + \beta}^3\) $5 \alpha^2 + \alpha^2 = 6 \alpha^2$ and $6 \alpha + 5 \alpha = 11 \alpha$
\(\ds \) \(=\) \(\ds \alpha^3 \paren {\alpha^4 + 4 \alpha^3 \beta + 6 \alpha^2 \beta^2 + 4 \alpha \beta^3 + \beta^4} + \alpha^3 \paren {\alpha^3 + 3\alpha^2 \beta + 3 \alpha \beta^2 + \beta^3}\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds 6 \alpha^2 \paren {\alpha^4 + 4 \alpha^3 \beta + 6 \alpha^2 \beta^2 + 4 \alpha \beta^3 + \beta^4} + 6 \alpha^2 \paren {\alpha^3 + 3\alpha^2 \beta + 3 \alpha \beta^2 + \beta^3}\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds 11 \alpha \paren {\alpha^4 + 4 \alpha^3 \beta + 6 \alpha^2 \beta^2 + 4 \alpha \beta^3 + \beta^4} + 11 \alpha \paren {\alpha^3 + 3\alpha^2 \beta + 3 \alpha \beta^2 + \beta^3}\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds 6 \paren {\alpha^4 + 4 \alpha^3 \beta + 6 \alpha^2 \beta^2 + 4 \alpha \beta^3 + \beta^4} + 6 \paren {\alpha^3 + 3\alpha^2 \beta + 3 \alpha \beta^2 + \beta^3}\)
\(\ds \) \(=\) \(\ds \alpha^7 + \paren {4 \beta + 1 + 6} \alpha^6 + \paren {6 \beta^2 + 3 \beta + 24 \beta + 6 + 11} \alpha^5 + \paren {4 \beta^3 + 3 \beta^2 + 36 \beta^2 + 18 \beta + 44 \beta + 11 + 6} \alpha^4\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds \paren {\beta^4 + \beta^3 + 24 \beta^3 + 18 \beta^2 + 66 \beta^2 + 33 \beta + 24 \beta + 6} \alpha^3 + \paren {6 \beta^4 + 6 \beta^3 + 44 \beta^3 + 33 \beta^2 + 36 \beta^2 + 18 \beta} \alpha^2 + \paren {11 \beta^4 + 11 \beta^3 + 24 \beta^3 + 18 \beta^2} \alpha + 6 \beta^4 + 6 \beta^3\)
\(\ds \) \(=\) \(\ds \alpha^7 + \paren {4 \beta + 7} \alpha^6 + \paren {6 \beta^2 + 27 \beta + 17} \alpha^5 + \paren {4 \beta^3 + 39 \beta^2 + 62 \beta + 17} \alpha^4 + \paren {\beta^4 + 25 \beta^3 + 84 \beta^2 + 57 \beta + 6} \alpha^3\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds \paren {6 \beta^4 + 50 \beta^3 + 69 \beta^2 + 18 \beta} \alpha^2 + \paren {11 \beta^4 + 35 \beta^3 + 18 \beta^2} \alpha + 6 \beta^4 + 6 \beta^3\)

$\Box$

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