When I learned probability, we were basically presented with a random variable X, told that it could occupy a bunch of different values, and asked to calculate what the average/expected value is based on the frequencies of what those different values could be. So you start with a question like “we roll a die. here are all the values it could be and they all happen one-sixth of the time. Add each value multiplied by one-sixth to each other to get the expected value.” This framing naturally leads to definition (1) when you expand to continuous random variables.
On one hand, this makes definition (1) really intuitive and easy to learn. After all, if you frame the questions around the target space, you’ll frame your understanding around the target space. Frankly, when I read your comment, my immediate reaction was “what on earth is a probability space? we’re just summing up the ways the target variable can happen and claiming that its a map from some other space to the target variable is just excessive!” When you’re taught about target space, you don’t think about probability space.
On the other hand, defintiion (2) is really useful in a lot of (usually more niche) areas. If you don’t contextualize X as a map between a space of possible outcomes as a real number, things like integrals using Maxwell Boltzmann statistics won’t make any sense. To someone who does, you’re just adding up all the possibilities weighted by a given value.
When I learned probability, we were basically presented with a random variable X, told that it could occupy a bunch of different values, and asked to calculate what the average/expected value is based on the frequencies of what those different values could be. So you start with a question like “we roll a die. here are all the values it could be and they all happen one-sixth of the time. Add each value multiplied by one-sixth to each other to get the expected value.” This framing naturally leads to definition (1) when you expand to continuous random variables.
That’s a strong steelman of the status quo in cases where random variables are introduced as you describe. I’ll concede that (1) is fine in this case. I’m not sure it applies to cases (lectures) where probability spaces are formally introduced – but maybe it does; maybe other people still don’t think of RVs as functions, even if that’s what they technically are.
When I learned probability, we were basically presented with a random variable X, told that it could occupy a bunch of different values, and asked to calculate what the average/expected value is based on the frequencies of what those different values could be. So you start with a question like “we roll a die. here are all the values it could be and they all happen one-sixth of the time. Add each value multiplied by one-sixth to each other to get the expected value.” This framing naturally leads to definition (1) when you expand to continuous random variables.
On one hand, this makes definition (1) really intuitive and easy to learn. After all, if you frame the questions around the target space, you’ll frame your understanding around the target space. Frankly, when I read your comment, my immediate reaction was “what on earth is a probability space? we’re just summing up the ways the target variable can happen and claiming that its a map from some other space to the target variable is just excessive!” When you’re taught about target space, you don’t think about probability space.
On the other hand, defintiion (2) is really useful in a lot of (usually more niche) areas. If you don’t contextualize X as a map between a space of possible outcomes as a real number, things like integrals using Maxwell Boltzmann statistics won’t make any sense. To someone who does, you’re just adding up all the possibilities weighted by a given value.
That’s a strong steelman of the status quo in cases where random variables are introduced as you describe. I’ll concede that (1) is fine in this case. I’m not sure it applies to cases (lectures) where probability spaces are formally introduced – but maybe it does; maybe other people still don’t think of RVs as functions, even if that’s what they technically are.