What is the meaning of (sampling) a continuous probability distribution?

Question

According to theory, a continuous probability function is $f:\mathbb R\rightarrow\mathbb R$, such that $f$ is continuous and the improper integral of it is $1$. Moreover the function is never negative. The concept makes sense in that these distributions can be thought of as a limit as discrete RVs are summed.

So far so good... But then the words "let $x_1,...,x_n$ be a sample from a contiuous probability distribution" are uttered. This to me sounds a bit mythical, I don't think there is any way we could sample a continuous distribution.

What actually is the appropriate meaning of such sentences?

Answer 1

There is no "appropriate meaning". You are missing nothing. This is not meant to have any exact real-world meaning but is instead a mathematical idealization.

The completely rigorous meaning from a purely mathematical perspective is that $x_1,\dots,x_n$ are random variables on some probability space which are independent and have the given continuous distribution function $f$. That is, for any $r_1,\dots,r_n\in\mathbb{R}$, the probability that $x_i<r_i$ is true for all $i=1,\dots,n$ is $\prod_{i=1}^n\int_{-\infty}^{r_i} f(x)dx$.

Answer 2

A continuous probability distribution has continuous random variables. The probability density function, pdf, f(x) follows some rules.

$$ f(x) \geq 0 $$

$f(x) $ is piece wise continuous and

$$\int_{-\infty}^{\infty} f(x) dx = 1 $$

Also if $X$ is a continuous random variable falling in the interval $(a,b)$

$$ Pr(a < X < b) = \int_{a}^{b} f(x) dx$$

Importantly

$$ Pr(X=c) = \int_{c}^{c} f(x) dx = 0 $$