In the book Statistical Inference by George Casella, it is written that

An experimenter uses the information in a sample $X_1,X_2,\ldots,X_n$ to make inferences about an unknown parameter $\theta$.

I have always read "Suppose $X$ is a random variable and we take a sample $x_1,x_2,\ldots,x_n$." That is, $X$ can take some values randomly and $x_1,x_2,\ldots,x_n$ are some observed values of $X$. Capital letter $X$ denotes random variable and small letter $x$ denotes a realization from $X$.

But in the book mentioned above it is written "sample $X_1,X_2,\ldots,X_n$", which seems to me if I have $N$ random variables, then I took $n$ random variables randomly from $N$ random variables. But different random variables cannot infer one unknown parameter $\theta$.

  • How can I explain "sample $X_1,X_2,\ldots,X_n$" by real life example?

Someone tried to give me the example that suppose you want to estimate average height of students in your class. You took 10 students randomly. Now before observed, your height itself a random variable. But I have not understood "Is the statement true that your height itself a random variable?". When my height is observable (not observed) one can think of my height as 5.5', 5.6', 5.4', etc. But when it is observed it becomes fixed. It is not clear to me that my height itself a random variable.

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    $\begingroup$ As aficionados of this site will know, I love the "tickets in a box" model of random variables. It is a rigorous metaphor for how all probability models work: you write all possible outcomes on some tickets, put them into a box in proportions according to their underlying probabilities, and mix them up. Your height is the value on one particular ticket: it is not random. A random variable is the process of blindly drawing one ticket. A realization is the value on a ticket that has been drawn in this way. $\endgroup$ – whuber Mar 12 '15 at 13:58
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    $\begingroup$ In your "height"-example, $X_i$ would be defined as "Height of i-th randomly picked student". All these $X_i$ can be viewed as independent copies of the random variable $X$ defined as "Height of randomly picked student". $\endgroup$ – Michael M Mar 12 '15 at 14:50

Here is a quick summary of the way I (try to) explain this to my students (who are unfortunately not fluent in mathematics...).

  • a random experiment is an experiment which, when repeated, can produce different outputs.

  • such an experiment can be, e.g.: pick a random person in Paris and measure them. The output is a height. Each particular experiment leads a height $x$, which is a real number. We denote $X$ the random variable which is associated to this experiment; $X$ is not a number. A way to see it is as function from the set $\Omega$ of all possible experiment results, in our case the set of Parisians, to the real numbers. In this optic, a random variable is a measure done on a random experiment.

  • if you repeat this experiment, say, 10 times, you’ll get 10 sizes. But the day after, you do this again: pick 10 random guys, measure them. You’ll get 10 other heights. You can repeat this over and over, and you’ll get a bunch of different results. This is a (new) random experiment. The set of possible outcomes is the set of all 10-tuples of Parisians. For each of these experiments, you have 10 measures, that is 10 random variables $X_1, \dots, X_{10}$. This also explain that the arithmetic mean ${1\over 10}(X_1 + \cdots + X_{10})$ is a random variable, to be studied as such.

I hope this helps a bit. I swear that when I make the gestures, this is somehow convincing.

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    $\begingroup$ In case you're still here. Really nice answer (+1, btw), there is a point in your answer that really struggles me. "Each particular experiment leads a size 𝑥, which is a real number. We denote 𝑋 the random variable which is associated". Given that $\Omega$ is the sample space (in that case the set of all Parisians), how is $X: \Omega \to \mathbb{R}$ defined? Is for example, constant on all $\Omega$, i.e., $X(\omega)=x$? If not how would you write it? If you find some time please, give this a try because I've spent so much time to find a convincing answer. Thanks. $\endgroup$ – user430191 Mar 27 at 17:58
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    $\begingroup$ If $\omega$ is a Parisian, $X(\omega)$ is its height! $\endgroup$ – Elvis Mar 31 at 12:24
  • $\begingroup$ thanks for the response. So a random variable $X$ associated to a given $x \in \mathbb{R}$, is any variable $X: \Omega \to \mathbb{R}$ such that $X(\omega)=x$? I'm confused by "the random variable associated". Associated to what exactly do you mean? $\endgroup$ – user430191 Mar 31 at 13:03
  • $\begingroup$ If $\Omega$ is the set of all Parisians, for any guy $\omega\in\Omega$, $X(\omega)$ is his height. It depends on the guy you pick in $\Omega$, so it is a function from $\Omega$ to $\mathbb R$. This is called a random variable because you specify a probability distribution on $\Omega$ (usually the uniform distribution). Its a a associated to the random experiment: "pick up a random guy and measure him". In a concrete random experiment, you pick a guy, let's denote him $\omega_1$, and measure him: you get a height, that you can denote $x_1 = X(\omega_1) \in\mathbb R$. $\endgroup$ – Elvis Mar 31 at 13:25
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    $\begingroup$ thanks a lot @Elvis, that demystifies many things $\endgroup$ – user430191 Mar 31 at 13:59

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