Mean of Sum of IID Random Variables

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SUMMARY

The mean of a sum of independent identically distributed (IID) random variables is equal to the sum of their means. Specifically, if Y is defined as Y = X1 + X2 + ... + Xn, then the expected value E[Y] is equal to nE[X], where E[X] is the expected value of a single random variable X. This principle holds true regardless of whether the random variables are independent or have the same distribution, confirming that the mean of a sum is indeed the sum of the means.

PREREQUISITES
  • Understanding of random variables (RVs)
  • Knowledge of expected value (E[X])
  • Familiarity with independent and identically distributed (IID) concepts
  • Basic principles of probability theory
NEXT STEPS
  • Study the properties of expected values in probability theory
  • Explore the Central Limit Theorem and its implications for sums of random variables
  • Learn about variance and how it relates to sums of random variables
  • Investigate applications of IID random variables in statistical modeling
USEFUL FOR

Statisticians, data scientists, and anyone involved in probability theory or statistical analysis will benefit from reading this discussion.

ObliviousSage
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If X is some RV, and Y is a sum of n independent Xis (i.e. n independent identically distributed random variables with distribution X), is the mean of Y just the sum of the means of the n Xs?

That is, if Y=X1+X2+...+Xn, is E[Y]=nE[X]?

I know that for one-to-one order-preserving functions, if Y=h(X) then E[Y]=E[h(X)] with a single variable X, but I'm not sure if it works with multiple Xs, even with something as simple as addition.
 
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The mean of a sum is the sum of the means. The terms in the sum do not have to be independent or have the same distribution.
 
mathman said:
The mean of a sum is the sum of the means. The terms in the sum do not have to be independent or have the same distribution.

Awesome, I wasn't sure. Thanks for clearing that up!
 

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