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I am doing a problem that reads

Suppose $X_1, X_2..., X_n$ are independent random variables with common expectation $\mu$ and variance $\sigma^2$. Let $S_n$=$X_1+X_2+...+X_n$. Find the expectation and variance of $S_n$ repeat for $S_n/n$.

I am kind of at a loss. I assumed the expectation would remain the same for $S_n$, if I have two random variables that can take on the same values and have the same expectation, wouldn't the expectation remain the same because I would be doubling both the numerator and the denominator?

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The expectation is a linear operator: $$\mathbb{E}[\alpha X+ \beta Y] = \alpha\mathbb{E}[X]+ \beta\mathbb{E}[Y] $$ for any random variables $X,Y$ (not necessarily independent) that have well-defined expectations; and constants $\alpha,\beta$.

The variance is not, but you have: $$ \operatorname{Var}(\alpha X) = \alpha^2\operatorname{Var}(X) $$ for any random variable $X$ that has a well-defined variance, and constant $\alpha$. Moreover, if both $X,Y$ have a variance, and $X,Y$ are independent, then $$ \operatorname{Var}(X+Y) = \operatorname{Var}(X) + \operatorname{Var}(Y). $$

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  • $\begingroup$ Thanks, but I am not so familiar with linear operators. This is the first time I have encountered the term. I will endeavour to learn more about this, but is there a way to explain without reference to this concept? $\endgroup$ – user183974 Oct 12 '15 at 14:52
  • $\begingroup$ You can just forget the word "linear" and use the property I wrote as a given. Basically, it follows from the fact that expectation is an integral: $\int (\alpha x f_X(x) + \beta x f_Y(x))dx = \alpha \int x f_X(x) dx + \beta \int x f_Y(x) dx $. $\endgroup$ – Clement C. Oct 12 '15 at 14:54

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