The mean of a sum of variables.

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Discussion Overview

The discussion centers on proving that the mean of a random variable Z, defined as the sum of two other random variables X and Y, equals the sum of the means of X and Y. The scope includes theoretical aspects of probability and integration related to random variables.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant asks how to prove that the mean of Z (X + Y) equals the sum of the means of X and Y.
  • Another participant suggests that this theorem is analogous to the property that the integral of a sum is the sum of the integrals.
  • A participant expresses difficulty in relating the integral property to the mean of the sum of random variables and mentions encountering a convoluted integral.
  • It is proposed that a proof can be established for independent random variables using a change of variables and reversing the order of integration.
  • For dependent variables, a suggestion is made to consider the joint and marginal distributions of X and Y.
  • One participant provides a mathematical expression for the expected value of the sum of two random variables, indicating how to separate the integrals for X and Y.

Areas of Agreement / Disagreement

Participants express varying degrees of understanding and ability to prove the theorem, with some uncertainty regarding the application to dependent versus independent variables. No consensus is reached on a definitive proof.

Contextual Notes

Some participants mention challenges with convoluted integrals and the need for specific proofs, indicating potential limitations in their current understanding or approach.

Mppl
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How do I prove that the mean of a random variable Z which is the sum of to other random variables X and Y is the sum of the mean of X with the mean of Y?
 
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Essentially the theorem is equivalent to the theorem that the integral of a sum is the sum of the integrals.
 
well I obviously know that the integral of the sum is the sum of the integral but I don't know how I can relate that to the situation a mentioned, can you please be more specific?
I'm trying to prove it and I'm getting a convultion integral so far...
thank you.
 
Mppl said:
well I obviously know that the integral of the sum is the sum of the integral but I don't know how I can relate that to the situation a mentioned, can you please be more specific?
I'm trying to prove it and I'm getting a convultion integral so far...
thank you.

Yes that'll eventually give you a proof for the special case where the rv's are independent and have densities (involves reversing the order of integration and a change of variables).

Another approach that would work for the non-independent case is to consider separately the joint distribution and marginal distributions of X and Y.
 
well but I'm not being able to prove it either for dependent or independent variables, can you please show me the proof or tell me where I can find it? thank you
 
Two random variables.
E(X+Y)=∫∫(x+y)dF(x,y)=∫∫xdF(x,y) + ∫∫ydF(x,y).
Integrate with respect to y in the first integral and integrate with respect to x in the second integral. You will be left with E(X) + E(Y).

In the above F(x,y) is the joint distribution function.
 

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