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In Statistics, it is very common to see i.i.d random variables $\{X_1, X_2,...,X_n \}$. I wonder whether or not it is assumed that these random variables come from the same probability space $(\Omega, \mathcal{F}, P)$, and can you construct i.i.d. random variables from the same probability space?

Another question is that in introductory probability course, you are often asked to compute the probability of the random variable $X$ equal to the random variable $Y$. For example, $X,Y$ are i.i.d random variables distributed according to $Ber(\theta)$. Do $X$ and $Y$ have to come from the same probability space $(\Omega, \mathcal{F}, P)$ or not?

Is a new probability space simply assigned whenever a new experiment is conducted? (even if the distributions of random variables are the same in the case of i.i.d random variables)

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    $\begingroup$ The sample space $\Omega$ can be very large when you have $n$ random variables, and is at least all as big as the possible combinations of outcomes of the different random variables. In your second paragraph, the measurable event $X=0 \cap Y=1$ is in effect one of the elements of $\mathcal F$, as is the event $X=0$ and the event $Y=1$; so yes, they come from the same probability space $(\Omega, \mathcal{F}, P)$ $\endgroup$
    – Henry
    Jun 15, 2022 at 12:46

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Henry's answer is good. For the first question, I wanted to point you to the Kolmogorov extension theorem, which gives sufficient conditions under which you can put a collection of random variables into the same probability space. This theorem is good-enough for most situations in introductory probability.

For the second question, certainly you need $X,Y$ in the same probability space to ask for the probability of the event $\{X=Y\}$ (which is shorthand for $\{\omega \in \Omega \, : \, X(\omega) = Y(\omega) \}$).

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