# Confused about notation on "All of Statistics" bootstrap

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!

• 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. Jun 18, 2020 at 21:15
• 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. Jun 19, 2020 at 17:36

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.