- #1

etotheipi

I've read that we can define a random variable on a probability space ##(\Omega, F, P)## such that it is a function that maps elements of the sample space to a measurable space - for instance, the reals - i.e. ##X: \Omega \rightarrow \mathbb{R}##.

That being said, it's often treated (at least from what I've encountered in elementary probability) as a variable, whose value becomes known after the experiment has been completed. When we write the probability of a certain event, like ##P(X<5)##, then I guess what we really mean is ##P(X(\omega)<5)## where ##\omega \in \Omega##. But if - as if often the case - the elements of ##\Omega## are also real numbers, then a perfectly good mapping would be ##X(\omega) = \omega##, so we need not even think too much about it.

My question is, is it correct to think of the random variable as analogous to a variable like velocity, which happens to also be a function (of time, or other parameters)? As in, we may define velocity as a function, ##v(t) = \frac{dx}{dt}##, but more often than not I tend to think of ##v## as a variable. Can we treat a random variable ##X## in the same way, as a variable?

That's what the name seems to imply, but names can be misleading... the text I'm using makes a point to state that it is definitely not a variable (nor is it random)!

That being said, it's often treated (at least from what I've encountered in elementary probability) as a variable, whose value becomes known after the experiment has been completed. When we write the probability of a certain event, like ##P(X<5)##, then I guess what we really mean is ##P(X(\omega)<5)## where ##\omega \in \Omega##. But if - as if often the case - the elements of ##\Omega## are also real numbers, then a perfectly good mapping would be ##X(\omega) = \omega##, so we need not even think too much about it.

My question is, is it correct to think of the random variable as analogous to a variable like velocity, which happens to also be a function (of time, or other parameters)? As in, we may define velocity as a function, ##v(t) = \frac{dx}{dt}##, but more often than not I tend to think of ##v## as a variable. Can we treat a random variable ##X## in the same way, as a variable?

That's what the name seems to imply, but names can be misleading... the text I'm using makes a point to state that it is definitely not a variable (nor is it random)!

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