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Background

This is related to the following question: Is an experiment's Sample Space always useful?

I have been reading about Sample Spaces, and I understand they represent all possible outcomes of an experiment, and that a random variable is like a function over the sample space.

In the answer to my previous question, I was told that using a Random Variable for the sample space makes it easier to solve complicated problems. But I don't understand why using a Random Variable makes it easier or what problems it makes it easier for.

To be clear

I understand what a Random Variable is. What I don't understand is why it's more helpful than a sample space in complex problems.

Question

What is it about Random Variables that makes them more useful than sample spaces on complex problems?

Connor
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  • Is this a duplicate? The linked question is on the meaning of a random variable. That's clearly not what's being asked here...? – Connor Dec 11 '23 at 15:39
  • The answers in the duplicate thread also answer your question here -- or, rather, show how your question doesn't make sense. It's not a matter of "more useful," but how the concepts are applied. – whuber Dec 11 '23 at 15:40
  • Wouldn't it be better to have a clear question and answer? I'm reading through the question now and it doesn't clearly answer my question at all. – Connor Dec 11 '23 at 15:42
  • Here's an analogy: what is it about buildings that makes them more useful than roads? It's clear but unhelpful because it presupposes some hierarchy of usefulness pertains. It cannot lead to the kind of clear, well-supported, unambiguous answers we seek here. – whuber Dec 11 '23 at 15:45
  • I've read through your answer on the linked duplicate. It's a very nice overview of a random variable, one of the best I've read, so thank you for that. But it's not clear from that answer why it's easier to use them in complex problems. Perhaps I should change this question to a "when" question if you have a problem with me saying de-facto that random variables are more useful than sample spaces? – Connor Dec 11 '23 at 15:49
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    "Easier" than what alternative? Sample spaces are not alternatives to random variables, just as roads are not alternatives to buildings. Each has its own distinct role in statistics or infrastructure, respectively. Thank you for the compliments, btw. – whuber Dec 11 '23 at 15:52
  • Perhaps this is showing my newness to statistics, I'll try and explain. I understand that a random variable is based on a sample space. That they're connected. But that a sample space can still be inherently useful by itself. For example, in simple dice problems you can count outcomes and find probabilities without introducing the idea of a random variable. Now in this question: https://stats.stackexchange.com/questions/633523/is-an-experiments-sample-space-always-useful I asked if the sample space is always useful and was told in complex problems random variables are used instead. – Connor Dec 11 '23 at 15:57
  • I want to know why they're used instead. What is it about them that makes them easier to use in a problem that has a difficult-to-define sample space? Or an infinite sample space, etc. – Connor Dec 11 '23 at 15:58
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    @Connor Perhaps, the question should be, "what is an example when using random variables simpler than using sample spaces". – Nicolas Bourbaki Dec 11 '23 at 16:13

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