# Skewness and Kurtosis of Bernoulli Distributions

Suppose you have multiple independent Bernoulli random variables, $$X_1,X_2,...,X_n$$, with respective probabilities of success $$p_1,p_2,...,p_n$$.

So $$E(X_i)=p_i$$, and $$E(X_i+X_j)=E(X_i)+E(X_j)$$. Also, $$\text{var}(X_i)=p\cdot (1-p)$$, and $$\text{var}(X_i+X_j)=\text{var}(X_i)+\text{var}(X_j)$$. (Though correct me if any of that is wrong.)

I'm trying to figure out the similar formulae for skewness and kurtosis. Since skewness of $$X_i$$ is given by $$\tfrac{(1-p_i)-p_i}{\sqrt{p_i\cdot(1-p_i)}}$$, how would you calculate the skewness of $$X_i+X_j$$? And for kurtosis of $$X_i+X_j$$?