Excess Kurtosis of Beta Distribution/Lemma 4

Lemma for Excess Kurtosis of Beta Distribution

\(\ds 3 \alpha^3 \paren {\alpha + \beta + 1}^2 \paren {\alpha + \beta + 2} \paren {\alpha + \beta + 3}\)

\(=\)

\(\ds 3 \alpha^7 + \paren {12 \beta + 21} \alpha^6 + \paren {18 \beta^2 + 63 \beta + 51} \alpha^5 + \paren {12 \beta^3 + 63 \beta^2 + 102 \beta + 51} \alpha^4 + \paren {3 \beta^4 + 21 \beta^3 + 51 \beta^2 + 51 \beta + 18} \alpha^3\)

Proof

\(\ds 3 \alpha^3 \paren {\alpha + \beta + 1}^2 \paren {\alpha + \beta + 2} \paren {\alpha + \beta + 3}\)

\(=\)

\(\ds 3 \alpha^3 \paren {\alpha + \beta + 1}^2 \paren {\paren {\alpha + \beta }^2 + 5 \paren {\alpha + \beta } + 6}\)

Multiply $\paren {\alpha + \beta + 2} \paren {\alpha + \beta + 3}$

\(\ds \)

\(=\)

\(\ds 3 \alpha^3 \paren {\paren {\alpha + \beta }^2 + 2 \paren {\alpha + \beta } + 1} \paren {\paren {\alpha + \beta }^2 + 5 \paren {\alpha + \beta } + 6}\)

Expand $\paren {\alpha + \beta + 1}^2$

\(\ds \)

\(=\)

\(\ds 3 \alpha^3 \paren {\paren {\alpha + \beta }^4 + 5 \paren {\alpha + \beta }^3 + 6 \paren {\alpha + \beta }^2 + 2 \paren {\alpha + \beta }^3 + 10 \paren {\alpha + \beta }^2 + 12 \paren {\alpha + \beta } + \paren {\alpha + \beta }^2 + 5 \paren {\alpha + \beta } + 6 }\)

\(\ds \)

\(=\)

\(\ds 3 \alpha^3 \paren {\paren {\alpha + \beta }^4 + 7 \paren {\alpha + \beta }^3 + 17 \paren {\alpha + \beta }^2 + 17 \paren {\alpha + \beta } + 6 }\)

Rewriting

\(\ds \)

\(=\)

\(\ds 3 \alpha^3 \paren {\paren {\alpha^4 + 4 \alpha^3 \beta + 6 \alpha^2 \beta^2 + 4 \alpha \beta^3 + \beta^4} + 7 \paren {\alpha^3 + 3\alpha^2 \beta + 3 \alpha \beta^2 + \beta^3} + 17 \paren {\alpha^2 + 2 \alpha \beta + \beta^2} + 17 \paren {\alpha + \beta } + 6 }\)

\(\ds \)

\(=\)

\(\ds 3 \alpha^7 + \paren {12 \beta + 21} \alpha^6 + \paren {18 \beta^2 + 63 \beta + 51} \alpha^5 + \paren {12 \beta^3 + 63 \beta^2 + 102 \beta + 51} \alpha^4 + \paren {3 \beta^4 + 21 \beta^3 + 51 \beta^2 + 51 \beta + 18} \alpha^3\)

$\Box$