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I have some distributions with the same distribution, for example, Gaussian distribution or Beta distribution. My question: is the mean of the sum of these distributions equals to the sum of the means of these distributions?

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Yes. Mostly. By the property called linearity of expectation, the mean of the sum of random variables is the same as the sum of their individual means.

This is true so long as there are finitely many random variables going into the sum and so long as their means are finite. Also true in some other cases, see comment below.

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  • $\begingroup$ IMHO, you cannot say this for all situations: According to Fubini-Tonelli theorem, there are some conditions to be met for this interchange. See, for example, MITZENMACHER 2005 Probability and Computing: randomized Algorithms and Probabilistic Analysis, page 23. $\endgroup$ Aug 17, 2019 at 14:00
  • $\begingroup$ @ErdoganCEVHER Thank you for your comment, I have expanded a bit. $\endgroup$
    – einar
    Aug 19, 2019 at 7:41

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