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## Main Question or Discussion Point

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

So [tex]E(X_i)=p_i[/tex], and [tex]E(X_i+X_j)=E(X_i)+E(X_j)[/tex]. Also, [tex]\text{var}(X_i)=p\cdot (1-p)[/tex], and [tex]\text{var}(X_i+X_j)=\text{var}(X_i)+\text{var}(X_j)[/tex]. (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 [tex]X_i[/tex] is given by [tex]\tfrac{(1-p_i)-p_i}{\sqrt{p_i\cdot(1-p_i)}}[/tex], how would you calculate the skewness of [tex]X_i+X_j[/tex]? And for kurtosis of [tex]X_i+X_j[/tex]?

So [tex]E(X_i)=p_i[/tex], and [tex]E(X_i+X_j)=E(X_i)+E(X_j)[/tex]. Also, [tex]\text{var}(X_i)=p\cdot (1-p)[/tex], and [tex]\text{var}(X_i+X_j)=\text{var}(X_i)+\text{var}(X_j)[/tex]. (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 [tex]X_i[/tex] is given by [tex]\tfrac{(1-p_i)-p_i}{\sqrt{p_i\cdot(1-p_i)}}[/tex], how would you calculate the skewness of [tex]X_i+X_j[/tex]? And for kurtosis of [tex]X_i+X_j[/tex]?