If I have a set of indepenent and identically distributed random variables X1,...Xn, then [tex]Var(\sum_{i=1}^{n}X_i) = \sum_{i=1}^{n}Var(X_i)[/tex].(adsbygoogle = window.adsbygoogle || []).push({});

Now I want to know what the sum of variances of Xi would be when n is a random variable?

I'm guessing the above statement still holds when n is a random variable, but when I work out both sides of the above statement, I get two different answers.

For example, [tex]Var(\sum_{i=1}^{n}X_i)[/tex] will now be the variance of a random sum of random variables which can be worked out using the total law of variance, and comes out as E(n)*Var(X1) + Var(n)*E(X1)^2.

But evaluating the other side of the expression [tex]\sum_{i=1}^{n}Var(X_i)[/tex] when n is a random variable comes out as E(n)*Var(X1).

So, I don't understand why I'm getting two different answers here?? Which one is correct?? I think they should be the same.

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# Variance of a sum

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