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 independent*if, 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