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

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SUMMARY

The discussion confirms that for two independent random variables X and Y with an expectation of 0, the equation E((X+Y)^3) = E(X^3) + E(Y^3) holds true. This conclusion is derived through the application of the linearity of expectation and the independence of the variables. By expanding (X+Y)^3 and utilizing properties of expectation, it is established that terms like E[X^2Y] and E[XY^2] equal zero, reinforcing the validity of the equation.

PREREQUISITES
  • Understanding of linearity of expectation
  • Knowledge of independent random variables
  • Familiarity with the properties of expected values
  • Basic algebraic expansion techniques
NEXT STEPS
  • Study the concept of linearity of expectation in more depth
  • Explore the implications of independence in probability theory
  • Learn about higher moments of random variables
  • Investigate applications of moment generating functions
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Mathematicians, statisticians, and students of probability theory seeking to deepen their understanding of the properties of independent random variables and their expectations.

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TL;DR
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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