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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