Let $X \thicksim B(n,p)$ then I would like to evaluate kurtosis and skweness of X.
First I want to use the fact that kurtosis $k_3(\dfrac{X-\mu}{ \sigma})=\dfrac{k_3(X)}{\sigma^3}$ and skewness kurtosis $k_4(\dfrac{X-\mu}{ \sigma})=\dfrac{k_4(X)}{\sigma^4}$.
To use above identity, one needs to derive 3 and 4-th cumulant of X.
mgf of bionomial $X$ is $ M(t)=[(1-p)+pe^t]^n$ thus
$K(t) = \log M(t)=n\log [(1-p)+pe^t]$
My question is here:
How could one expand above log term into the form such as $\sum_{r=1}^{\infty}\dfrac{k_r(0)}{r!}t^r$?
textbook has denoted
$n\log [(1-p)+pe^t] = \sum_{r=1}^{\infty}\dfrac{t^r}{r!}\{n\sum_{k=1}^r\sum_{j_1+j_2...+j_k=r}\begin{pmatrix}r\\j_!,j_2...j_k\end{pmatrix}\dfrac{(-1)^{k-1}}{k}p^k\}$
How could above identity be derived?