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Let $X$ denote a random variable. Then from a rigorous mathematical perspective (books such as Durrett, Feller, Kolmogorov, etc.),

$X$ is a function. $X: \Omega \to \mathbb{R}^n$.

Domain of the function is the sample space, $\Omega$

Range is a value in $\mathbb{R}^n$.


In supervised learning, let $X, Y$ denote the random variable corresponding to the data $x_n \in \mathbb{R}^n$ and target $y_n \in \mathbb{R}$ respectively.

Then $X$ maps from a sample space into a piece of data $x_n$. And $Y$ maps from a sample space into a piece of target, $y_n$.

With this, the notation, such as $p_{X|Y}(x_n|y_n) = \Pr[X = x_n| Y = y_n]$

is completely well defined.

Then we can start talking about things like logistic regression, i.e.,

$p_{X|Y}(x_n|y_n) = \Pr[X = x_n| Y = y_n] = \text{logit}(y_nw^Tx_n) \in (0,1)$

etc.

But what is the sample space $\Omega$? Any example would be much appreciated!

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