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I have a problem understanding the difference between random variable and random sample. I have read this thread, but still it is unclear.

According to wiki random variable is a function that from a set of outcomes (events) to measurable values $\Omega$ ->$E$ where $E$ could be $\mathbb{R}$. And random sample is a possible outcome.

So, in a lecture script I have these sentences which are confusing me:

let $D=\{x_1,x_2,x_3...x_n\}$ be a set of random variables.

Now I want to create a concrete example for myself to understand what $\{x_1,x_2,...,x_n\}$ could be.

Let's take the sample of two dice where we want to compute the probability of the sum of the figures of dice we have thrown. So we construct a random variable $X$ which just adds the thrown number of the dice:

$$X:\Omega\to\mathbb{N}$$ $$X(\text{die}_1,\text{die}_2) = \text{die}_1 + \text{die}_2 $$

Where dice are uniformly distributed in $ \{ 1,\cdots,6 \} $.

And as far I know, if I throw two diсe, I have concrete values which are my random samples.

Now my concrete question is: what could be in this specific example a set of random variables $D=\{x_1,x_2,x_3...x_n\}$? And what are then the concrete random samples?

UPDATE:
After reading the comments and replies, I want to underline, that I have a specific question and I would appreciate if someone could answer it explicitly: In the above example please give me a concrete $D = {x_1,x_2,...,x_n}$ as a set of random variables. I want to see how a set of random variables $D = {x_1,x_2,...,x_n}$ (more than 2 random variables) would look like in this example.

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    $\begingroup$ It is important to be aware of the fact that there is a reality and a probabilistic model of the reality. The probabilistic model is usually a probability space $(\Omega,\mathcal A, P)$. On this space we can define random variables $X_1,\dots,X_n$ and "taking a random sample" can be interpreted as picking out some outcome $\omega\in\Omega$ (according to rules that are determined by the probability measure $P$) and then having a look at the values $X_1(\omega),\dots,X_n(\omega)$. The reality that corresponds with this picking out can be for instance throwing $n$ times a die. $\endgroup$
    – drhab
    Jul 16, 2018 at 19:04
  • $\begingroup$ @drhab So in this example a set $D=x_1,x_2...x_n$ would be the outcome of throwing n times a die? $\endgroup$
    – Code Pope
    Jul 16, 2018 at 22:39
  • $\begingroup$ In your example I would go for $\Omega=\{(i,j)\mid i,j\in\{1,2,3,4,5,6\}\}$ and $X_1,X_2$ prescribed respecively by $(i,j)\mapsto i$ and $(i,j)\mapsto j$. Then you have the random variables $X_1,X_2$ and every outcome $(i,j)$ induces the result of taking a sample: $(X_1(i,j),X_2(i,j))=(i,j)$. It might look redundant because in this simplistic example we get $(X_1,X_2)(\omega)=\omega$ so the outcome turns up again, but in more complicated situation with lots of data that will be different. Our interest is not in single outcomes but in events like $\{\omega\in\Omega\mid X(\omega)\in B\}$. $\endgroup$
    – drhab
    Jul 17, 2018 at 8:43

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This Three Things are totally different let's talk about them one by one

1.Random Variable: Suppose 'x' is a random variable so it can take any value from a positive number system like 1,2,3,4,5....... and so on

There are two types of random variables:

1.discret: which takes discrete value or we can say exact no like 1,2,3,4...

2.continuous: Which Takes Values like from 1 to 10 so we can say it may be 1.1111 or 3.444 or maybe 9.867545 anything

2.Data: Data Means Collection of lots of variables which has some labels like data of maturity in different ages

age-group: [1-10] [10-20]

maturity:-- [0.1] [0.2] So this is Data By which We can Get to the some specific result

3.Random Sample: Part Randomly Taken From Data We can say if we have a large amount of data we can not study on whole data then we take random part of data and them to study on that

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  • $\begingroup$ Can you please now give an example what a set of random variables $x_1,x_2,...x_n$ in my example is? $\endgroup$
    – Code Pope
    Jul 16, 2018 at 22:37
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    $\begingroup$ I would call this "the answer of a statistician" (so with strong focus on reality, and less on the probabilistic model that makes it possible to apply mathematics). $\endgroup$
    – drhab
    Jul 17, 2018 at 8:51
  • $\begingroup$ you can have distributions that are also part discrete and part continuous. $\endgroup$ Feb 4, 2023 at 20:19

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