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In Wikipedia's article on Sample there is the following remark:

''Note that a sample of random variables (i.e. a set of measurable functions) must not be confused with the realizations of these variables (which are the values that these random variables take). In other words, $X_i$ is a function representing the measurement at the $i$-th experiment and $x_i = X_i(ω)$ is the value we actually get when making the measurement.''

I'm afraid I don't understand this passage, can anybody please explain the point?

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4 Answers

up vote 3 down vote accepted

WP's formulation is at best ambiguous, in particular the second sentence quoted by the OP seems to hint at a distinction related to the precision of a measurement or to an approximation, or to intervals vs exact values. Nothing of the sort is pertinent. Rather, one wishes to distinguish a function from one of its values. For functions from $E$ to $F$, say, the first one is an element of $F^E$ and the second one is an element of $F$ (and in probability theory, $E$ is often denoted by $\Omega$ and $F$ could be $\mathbb{R}$ or a power of $\mathbb{R}$, but this is not important).

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Let's assume that there are infinitely many humans and that their IQ has a normal distribution with mean 100 and variance 15. Here is a possible experiment:

grab 10 people downtown and measure their IQ

As a mathematical model, we accept the setting that there is a probability space $\Omega$ and "grab human number i" is modelled by a "random variable" $X_i$ which is a measurable function $$ X_i: \; \Omega \to \mathbb{R} $$ such that the $X_i$ are independent and have a gaussian aka normal distribution.

When you go out and acutally perfom this experiment, you'll get ten humans and ten values (real numbers) for theiy IQ's. Let's call them $IQ_i$. With regard to our mathematical model, this means that for every human you convinced to do the IQ-Test there is a corresponding $\omega \in \Omega$, each of which is called an event, such that $$ IQ_i = X_i(\omega) $$

When you get out again and perform this experiment again, you can accept the same random variables $X_i$ as a model for your experiment, but you'll encounter different humans, get different results from the IQ-Tests, which corresponds to a different $\omega ' \in \Omega$.

Or, to put it shortly as Didier Piau did: The description of the experiment entails as a mathematical model the random variables aka measurable functions $X_i$; but everytime when you actually perform the experiment, it will result in a tuple of values of these functions.

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Shouldn't it be just $\omega$ instead of $\omega_i$? –  Rasmus Jun 25 '11 at 9:29
    
Right, thanks, I changed that. –  Tim van Beek Jun 25 '11 at 20:35
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I think that, as @Didier Piau pointed out, the message is not to confuse function values with the function itself. If this is so, the message seems quite trivial to me. It's like saying "don't confuse real numbers 4, 9, 16 with the function $f(x) = x^2$.'' Well, I believe nobody does!

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In casual discussion by non-experts, the confusion is in fact regrettably quite common. –  André Nicolas Jun 25 '11 at 16:58
    
@user How very true. –  Did Jun 26 '11 at 6:23
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When a random variable takes a value, you cannot always explicitly know this exact value, you can only measure it, i.e. use a function of the result which outputs you the measuring. For instance, would you be sampling real numbers in $[0,1]$ and measuring them only up to 3 digits of precision, if your first random variable $X_1$ gets value $\pi/4$ for instance, you could only say $X_1 = 0.785$, since the measurable function $X_1$ maps any $\omega$ in the interval $(0.7845,0.7855]$ to $0.785$.

In other words, $\omega$ is the "actual value", if we could say this, of the random variable and $X_1$ is a way to measure the actual value by letting it become a measurable function.

Hope that helps,

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So $X_1(\omega), X_2(\omega), ..., X_n(\omega)$ is the "true sample" and $x_1, x_2, ..., x_n$ is the "measured sample" where $x_i$ is an approximation of $X_i(\omega)$, right? –  Leo Jun 25 '11 at 4:46
    
@Patrick Da Silva: $\omega$ is not a value of the random variable, it ranges over the sample space. –  André Nicolas Jun 25 '11 at 5:28
    
@user : I know it is not the value... that's why I said "if we could say this". $\omega$ is "the value that happened" and the random variable is what we measure when this value happens. This is why $X_i$ is a function and not a value itself. –  Patrick Da Silva Jun 25 '11 at 6:34
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@Patrick: I disagree with your answer. The difference has nothing to do with precision or approximation, or interval vs exact value. Rather, one wishes to distinguish a function from one of its values. For functions from $E$ to $F$, say, one is an element of $F^E$ and the other an element of $F$ (and in probability theory, $E$ is often denoted $\Omega$ and $F$ could be $\mathbb{R}$ or a power of $\mathbb{R}$, but this is not important). –  Did Jun 25 '11 at 6:47
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@Patrick I also disagree, for the same reasons as Didier. In fact, if Didier were to reword his comment a bit it would make a good answer. –  Chris Taylor Jun 25 '11 at 8:39
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