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A population moment is often stated as

$E[x]$

the sample moment counterpart is stated as

$1/n \sum_{i=1}^{n} x_i$.

I do not quite understand what is meant by

$E[x]$.

Why not

$E[x_i]$?

What does

$E[x]$

say? Expected value of all observations of x? What is x?

Snoopy
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1 Answers1

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In short, $X$ is a random variable. It's not concrete value. It's like a representation of a random event. And yes, $E[X]$ is expected value of all observations of $X$.

If we talk about $E[X] = \frac{1}{n} \sum_{i=1}^{n} x_i$ - that's sample mean (or arithmetic mean). Sample mean is not "real" mean. It's like an approximation by the known value $\{x_1, \dots, x_n\} \in X$.

For example, for a continuous distribution of $X$ - $E[X] = \int_{-\infty}^{\infty} x f(x) dx$, where $f(x)$ is the probability density function.

renesat
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  • My question regards the method of moments in econometrics. I am asking the question from that perspective. What should be the correct notation for the population moment? That is, in the expected value, should we use $x$ or $x_i$? – Snoopy Dec 17 '23 at 21:57
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    @Snoopy. I think moments in econometrics have some logic. $X$ - random variable, and $x_i$ is the sample value, and $E[X]$ (usually not $E[X_i]$ if you not have many random variables) - expected value. It's common notation. And the wiki uses the same (but $W$ and $w_i$, not $x$): https://en.wikipedia.org/wiki/Method_of_moments_(statistics). – renesat Dec 17 '23 at 22:21