Independence & Conditional Independence
Two random variables, \(X\) and \( Y \) are independent if the occurance of one does not give any information on the likelihood of the other event occuring. In other words, their probability distribution can be expressed as a product of two factors, one only involving \(X\), and one only involving \(Y\):
\[\forall x \in X, y \in Y, \, p(X = x, Y= y) = p(X = x)p(Y = y)\]
Two random variables, x and y are conditionally independentif, given knowledge of the occurance of \(Z\), knowledge of the occurance of \(X\) provides no information on the likeihood of the occurance of \(Y\). This can be expressed as:
\[\forall x \in X, y \in Y, z \in Z, \, p(X = x, Y= y | Z = z) = p(X = x | Z = z)p(Y = y | Z = z)\]
Example
Independent events
x: the number on a rolled dice
y: the number on another rolled dice
Conditionally independent
x: a person's height
y: the person's vocabulary
z: the person's age