User:Herbmuell/sandbox

What?

 * where $I$ is the identity tensor, $ε(∇u) ≡ 1⁄2∇u + 1⁄2(∇u)^{T}$ is the rate-of-strain tensor and $∇ · u$ is the rate of expansion of the flow. So this decomposition can be explicited as:

Arfken.

Mathworld.

NIST.

OEIS.

The First Three Minutes Hello, please have a look here and leave your opinion if interested. Thank you. Herbmuell (talk) 12:01, 24 June 2018 (UTC)

Moments and properties
The first three raw moments are

\begin{align} \mu_1 & =\frac{n\alpha}{\alpha+\beta} \\[8pt] \mu_2 & =\frac{n\alpha[n(1+\alpha)+\beta]}{(\alpha+\beta)(1+\alpha+\beta)}\\[8pt] \mu_3 & =\frac{n\alpha[n^{2}(1+\alpha)(2+\alpha)+3n(1+\alpha)\beta+\beta(\beta-\alpha)]}{(\alpha+\beta)(1+\alpha+\beta)(2+\alpha+\beta)} \end{align} $$

and the kurtosis is

\gamma_2 = \frac{(\alpha + \beta)^2 (1+\alpha+\beta)}{n \alpha \beta( \alpha + \beta + 2)(\alpha + \beta + 3)(\alpha + \beta + n) } \left[ (\alpha + \beta)(\alpha + \beta - 1 + 6n) + 3 \alpha\beta(n - 2) + 6n^2 -\frac{3\alpha\beta n(6-n)}{\alpha + \beta} - \frac{18\alpha\beta n^{2}}{(\alpha+\beta)^2} \right]. $$

The cumulants are best expressed in terms of two alternative parameters:

$$\pi=\frac{\alpha}{\alpha+\beta} \!$$ is what?

$$ \rho= \tfrac{1}{\alpha+\beta+1}  \!$$ is the pairwise correlation between the n Bernoulli draws and is called the over-dispersion parameter.

Mean:

\mu = n\pi \!$$

Variance:

\sigma^2 = n\pi(1-\pi)[1+(n-1)\rho] \!$$

Skewness:

\gamma_1 = \left(1+\dfrac{2(N-1)R}{1+R}\right)\dfrac{1-2\bar{p}}{\sigma} \!$$

Excess kurtosis:

\gamma_2 =\left(1-6\bar{p}(1-\bar{p})+\dfrac{6NR(1+NR)}{(1+R)(1+2R)}\left[1-(5+R)\bar{p}(1-\bar{p})\right]\right)\dfrac{1}{\sigma^2} \!$$