I The third central moment of a sum of two independent random variables

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When X and Y are independent random variables, it is true that E((X+Y)^3) equals E(X^3) + E(Y^3) under the assumption that both have an expectation of 0. This conclusion is derived from the linearity of expectation and the properties of independent random variables. The expansion of (X+Y)^3 allows for the application of linearity, leading to terms like E[X^2Y] and E[XY^2] being equal to zero due to the independence and zero expectations of X and Y. Thus, the central moment of the sum of two independent random variables can be simplified effectively. The discussion confirms the validity of this mathematical relationship.
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Is it true that in probability theory the third central moment of a sum of two independent random variables is equal to the sum of the third central moments of the two separate variables?
Is it true that when X and Y are independent,

E ({X+Y}3) = E (X3)+E(Y3)?
 
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This is just linearity of the expectation. You are assuming X and Y have expectation 0 and are independent. Develop (X+Y)^3, use linearity of E[.], then use independence and centrality to get E[X^2Y] = E[X^2]E[Y]=0 and E[XY^2] = E[X]E[Y^2]=0.
 

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