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So I am currently reading "All of Statistics", and I am on the bootstrap chapter 8. I will transcribe a bit of the text to show what my confusion is.

Basically, when he says that $T_n = g(X_1, X_2, ..., X_n)$, is he saying that every data point is a random variable? I don't understand this, since each data-point that I have on a dataset is a vector right?

Also, why is it that if $T_n = \bar{X}$ then $\mathbb{V}_F(T_N) = \frac{\sigma^2}{n}$, and what is the meaning of $\int(x-\mu)^2 dF(x)$? I don't understand the meaning of $dF(x)$, I am only used to see this like $dx$ or $dy$. Thank you!

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    $\begingroup$ I was going to suggest you google Stieltjes integral, and you might want to do that, but the notation is not the same as in your text. $\endgroup$
    – BruceET
    Jun 18, 2020 at 21:15
  • $\begingroup$ In theoretical discussions, data values $X_i$ are often viewed as random variables. They have been sampled at random from a population. If the experiment were to be repeated, then all of the $X_i$s would likely change when a new random sample is chosen. // From the point of view of the experimenter, the data from the current experiment are fixed values to be described in a useful way: Perhaps to make a confidence interval for a population parameter; perhaps to test a hypothesis. But theoretical considerations show how best to summarize or describe data and how to draw inferences from them. $\endgroup$
    – BruceET
    Jun 19, 2020 at 17:36

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If $F$ is the CDF of a continuous population distribution, then $\int (x-\mu)^2\, dF(x)$ is $\sigma_X^2 = \int (x-\mu)^2 f_X(x)\, dx,$ where $f_X(x)$ is the density function of the distribution. (The integral is taken over the support of the distribution.)

If $F$ is the CDF of a discrete population distribution, then $\int (x-\mu)^2\, dF(x)$ is $\sigma_X^2 = \sum_x (x-\mu)^2 p_X(x)\, dx,$ where $p_X(x) = P(X = x),$ for each $x$ with positive probability.

The expression $\int (x-\mu)^2\, dF(x)$ can be used more generally for distributions that are neither continuous nor discrete. (A distribution that agrees with standard normal for positive $x$ and has $P(X = 0) = 1/2$ would be an example.)

Also, in your third paragraph, it's just saying that $V(\bar X) = \sigma_X^2/n,$ where $\bar X$ is the mean of a sample of size $n$ from a population with variance $\sigma_X^2,$ which I guess you already know.

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  • $\begingroup$ Thank you for this. What about regarding the notation on my first question? $\endgroup$
    – user272585
    Jun 19, 2020 at 8:50

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