2

In the stats book I am studying, while explaining bias and mean, the author goes like this:

"... Suppose a sample $$S=(X_1, X_2,...,X_n)$$ has been collected. ... Sample mean $\bar{X}$ estimates $\mu$ unbiasedly because its expectation is $$E(\bar{X})=E(\frac{X_1+...+X_n}{n})=\frac{EX_1+...+EX_n}{n}=\frac{n\mu}{n}=\mu $$."

What confuses me is this: The author treats each data item in the sample as if it is representing a different random variable $X_i$. How come? Why?

O. Altun
  • 235
  • 2
    You may be mentally conflating the idea of a random variable with the idea of the distribution of that random variable. It's likely (I'm guessing because you don't give sufficient context) that the $X_i$ all share the same distribution, but they're still distinct random variables. – Glen_b Dec 14 '14 at 21:36
  • 1
    To illustrate @Glen_b's point, if you suppose that a sample could be modeled as an $n$-tuple of the same random variable $X$, so that $S=(X,X,\ldots,X)$, then the only samples that would conform to this model would be those that consist of $n$ replicates of the same number! If you're good at critical reading, you might find it useful to spend some time with our thread on random variables. Although IMHO the accepted answer is worthless as a definition and some comments are confused, collectively the other answers might help your intuition. – whuber Dec 14 '14 at 21:48

0 Answers0