Reminder: This post contains 801 words
· 3 min read
· by Xianbin
Part I
We know that a random variable \(X\) is nothing but a mapping function from a space to a real number.
It is easy to think \(X\perp Y\) while the truth is not. We should be very careful when we claim independence. As Wiki explains, we say that \(X\perp Y \mid Z\) if and only if given \(Z\), \(X\) does not provide any information on \(Y\).
If we randomly partition a set \(V\) into disjoint \(X, Y\), and we randomly select an element \(a\in Y\), is the random variable \(a\) independent of \(X\)?
It is tempting to say that Yes. But it is wrong. But how to say that the event “choosing a is independent of \(X\)’’?
Now, if we only consider \(a\in Y\), that is, we know \(Y\). Then, apparently, we know \(a\) is one of the known \(Y\) and the information of \(X\) cannot help you gain any information about which one is \(a\). So, \(a\perp X \mid Y\).
Part II
For simplicity, we use \(P(A \mid x)\) to denote some conditional probability, which is short for \(P(A \mid X = x)\).