Mean of Sum of IID Random Variables

In summary, the mean of a sum is equal to the sum of the means of the individual random variables involved, regardless of their independence or distribution. This holds true even for multiple Xs, as long as they are one-to-one order-preserving functions.
  • #1
ObliviousSage
36
0
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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  • #2
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.
 
  • #3
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!
 

What is the "Mean of Sum of IID Random Variables"?

The "Mean of Sum of IID Random Variables" refers to the average value obtained when multiple independent and identically distributed (IID) random variables are added together.

How is the mean of sum of IID random variables calculated?

The mean of sum of IID random variables is calculated by taking the sum of the individual means of the random variables. This is a property of IID random variables, where the mean of the sum is equal to the sum of the means.

What does the mean of sum of IID random variables tell us?

The mean of sum of IID random variables gives us information about the expected value of the sum of the random variables. It can help us understand the overall behavior of the variables and make predictions about their combined outcomes.

Can the mean of sum of IID random variables be negative?

Yes, the mean of sum of IID random variables can be negative if the individual random variables have negative values. This means that the sum of the variables can result in a negative value.

How does the number of IID random variables affect the mean of sum?

The mean of sum of IID random variables increases as the number of variables increases, as long as they are independent and identically distributed. This is because the law of large numbers states that the average of a large number of IID random variables will converge to the expected value, which is the mean of sum of IID random variables.

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