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I am new in probability theory and still have some difficulties to capture the definition of random variables.

  • given repeated throws of dice will produce a sequence of random variables. A random variable being defined as a function on a measure space, the previous example could be modeled as being described with the aid of a single random variable: an outcome will be described by the (random) value of the function. But in fact it is not.

  • given a sample of a population for which the distribution of the height of each individual is to be determined is again thought of as each individual being a random variable and you are not considering a single random variable to describe the process.

It looks like a random variable assumes only one value. Can somebody explain what could my misunderstanding be ? Thanks.

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  • $\begingroup$ In the measure-theoretic framework, a random variable is a function on the sample space. Therefore given a value in the sample space, which might be called a "randomization parameter" and is traditionally denoted by $\omega$, you have a single real number (or vector, or whatever values the random variable takes). But $\omega$ is then randomly selected when we imagine actually performing the experiment. The value taken on by our random variable is also randomized in the process. $\endgroup$ – Ian Mar 8 '17 at 14:48
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Typically a random variable denotes a (real) number that is random. For example, let $X$ be the outcome of rolling a 6-sided die. Mathematically, we can model this by taking our probability space to be the unit interval and taking $X(\omega)$ to be the function $$ X(\omega)=\lfloor 6\omega\rfloor+1. $$ Then $\mathbb P(X=1)=\mathbb P(0\leq \omega<1/6)=1/6$, and similarly for the other values of $X$, so this truly models an unbiased die.

Now consider a sequence of $n$ dice rolls. Either we can consider each roll in isolation of the others, in which case we can use the same function $X(\omega)$ above to describe each roll, or we can consider the joint distribution of the dice rolls. Then the outcome of the experiment is a random vector $(X_1,\ldots,X_n)\in \mathbb R^n$. Thus we have a single random vector.

People also consider randomness in more general contexts, beyond just real numbers or real vectors. In this case it is common to use the term random element.

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The only confusion I'm seeing in your post regards terminology.

I've typically seen random variables used to refer to real-valued maps from a probability space, and random vectors to refer to vector-valued (e.g. $\mathbb{R}^n$) from a probability space. You can think of the realization of $n$ dice rolls as $n$ random variables (e.g. $X_1(\omega),...,X_n(\omega)$) or a single realization of a random vector (e.g. $f(\omega)=(X_1(\omega),...,X_n(\omega))$).

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