Excess Kurtosis of Beta Distribution/Lemma 3
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Lemma for Excess Kurtosis of Beta Distribution
| \(\ds 6 \alpha^2 \paren {\alpha + 1} \paren {\alpha + \beta} \paren {\alpha + \beta + 2} \paren {\alpha + \beta + 3} \paren {\alpha + \beta + 1}\) | \(=\) | \(\ds 6 \alpha^7 + \paren {24 \beta + 42} \alpha^6 + \paren {36 \beta^2 + 132 \beta + 102 } \alpha^5 + \paren {24 \beta^3 + 144 \beta^2 + 240 \beta + 102} \alpha^4\) | ||||||||||||
| \(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds \paren {6 \beta^4 + 60 \beta^3 + 174 \beta^2 + 168 \beta + 36} \alpha^3 + \paren {6 \beta^4 + 36 \beta^3 + 66 \beta^2 + 36 \beta} \alpha^2\) |
Proof
| \(\ds 6 \alpha^2 \paren {\alpha + 1} \paren {\alpha + \beta} \paren {\alpha + \beta + 2} \paren {\alpha + \beta + 3} \paren {\alpha + \beta + 1}\) | \(=\) | \(\ds 6 \alpha^2 \paren {\alpha + 1} \paren {\alpha + \beta} \paren {\alpha + \beta + 2} \paren {\paren {\alpha + \beta }^2 + 4 \paren {\alpha + \beta } + 3}\) | Multiply $\paren {\alpha + \beta + 3} \paren {\alpha + \beta + 1}$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds 6 \alpha^2 \paren {\alpha + 1} \paren {\alpha + \beta} \paren {\paren {\alpha + \beta }^3 + 4 \paren {\alpha + \beta }^2 + 3 \paren {\alpha + \beta } + 2 \paren {\alpha + \beta }^2 + 8 \paren {\alpha + \beta } + 6 }\) | Multiply by $\paren {\alpha + \beta + 2}$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds 6 \alpha^2 \paren {\alpha + 1} \paren {\alpha + \beta} \paren {\paren {\alpha + \beta }^3 + 6 \paren {\alpha + \beta }^2 + 11 \paren {\alpha + \beta } + 6 }\) | Rewriting | |||||||||||
| \(\ds \) | \(=\) | \(\ds 6 \alpha^2 \paren {\alpha + 1} \paren {\paren {\alpha + \beta }^4 + 6 \paren {\alpha + \beta }^3 + 11 \paren {\alpha + \beta }^2 + 6 \paren {\alpha + \beta } }\) | Multiply by $\paren {\alpha + \beta }$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds 6 \alpha^3 \paren {\alpha^4 + 4 \alpha^3 \beta + 6 \alpha^2 \beta^2 + 4 \alpha \beta^3 + \beta^4} + 36 \alpha^3 \paren {\alpha^3 + 3\alpha^2 \beta + 3 \alpha \beta^2 + \beta^3} + 66 \alpha^3 \paren {\alpha^2 + 2 \alpha \beta + \beta^2} + 36 \alpha^3 \paren {\alpha + \beta }\) | ||||||||||||
| \(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds 6 \alpha^2 \paren {\alpha^4 + 4 \alpha^3 \beta + 6 \alpha^2 \beta^2 + 4 \alpha \beta^3 + \beta^4} + 36 \alpha^2 \paren {\alpha^3 + 3\alpha^2 \beta + 3 \alpha \beta^2 + \beta^3} + 66 \alpha^2 \paren {\alpha^2 + 2 \alpha \beta + \beta^2} + 36 \alpha^2 \paren {\alpha + \beta }\) | |||||||||||
| \(\ds \) | \(=\) | \(\ds 6 \alpha^7 + \paren {24 \beta + 36 + 6} \alpha^6 + \paren {36 \beta^2 + 108 \beta + 66 + 24 \beta + 36 } \alpha^5 + \paren {24 \beta^3 + 108 \beta^2 + 132 \beta + 36 + 36 \beta^2 + 108 \beta + 66} \alpha^4\) | ||||||||||||
| \(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds \paren {6 \beta^4 + 36 \beta^3 + 66 \beta^2 + 36 \beta + 24 \beta^3 + 108 \beta^2 + 132 \beta + 36} \alpha^3 + \paren {6 \beta^4 + 36 \beta^3 + 66 \beta^2 + 36 \beta} \alpha^2\) | |||||||||||
| \(\ds \) | \(=\) | \(\ds 6 \alpha^7 + \paren {24 \beta + 42} \alpha^6 + \paren {36 \beta^2 + 132 \beta + 102 } \alpha^5 + \paren {24 \beta^3 + 144 \beta^2 + 240 \beta + 102} \alpha^4\) | ||||||||||||
| \(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds \paren {6 \beta^4 + 60 \beta^3 + 174 \beta^2 + 168 \beta + 36} \alpha^3 + \paren {6 \beta^4 + 36 \beta^3 + 66 \beta^2 + 36 \beta} \alpha^2\) |
$\Box$