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Skewness and Kurtosis of Bernoulli Distributions

  1. Mar 2, 2010 #1
    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]?
  2. jcsd
  3. Mar 3, 2010 #2


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    Skewness is a factor using the (centered) second and third moments, while kurtosis uses the (centered) second and fourth moments. You should be able to calculate the third and fourth moments and thus the quantities you want. You already have the centered second moment (variance).
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