I'm learning about things like the joint PDF of $n$ random variables. The typical way to do this is by multiplying the PDFs of the variables, assuming they are independent. However, why is it the case that we don't divide by the permutations of the random variables? i.e. we know that the event ${A\cap B}$ could have happened with $A$ first or $B$ first, so why don't we divide the resulting probability by $2$?
(I do understand that there's a constraint that the resulting joint PDF must integrate to 1, by definition, but that still doesn't make me comfortable with just leaving things as they are...some clarification would help. Thank you!)
i.e. Suppose $X_1,...,X_n$ is distributed independently with pdfs $f_1,...,f_n$. The resulting joint pdf is: $$ f(x_1,...,x_n) = \prod_{i=1}^{n} f_{i} $$ This makes sense, as we want the integral of the joint pdf to be 1: $$ \int_{x_1...x_n}\prod_{i=1}^{n} f_{i} = 1 $$ But, in other more elementary probability problems, we often divide by the number of permutations of an event, because we're interested in the probability of the unordered outcome.
Why don't we divide by the number of permutations here? i.e.
$$ f(x_1,...,x_n) = \frac{\prod_{i=1}^{n} f_{i}}{n!} $$
Of course, it doesn't integrate to 1, but I'm curious as to whether the order of events affects the meaning of the joint pdf and how it's constructed.