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

  • #1
Ad VanderVen
169
13
TL;DR Summary
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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  • #2
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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