🇺🇸 Independence

Two random variables and are independent if and only if

Intuitively, this is saying that if we know , does not change. Using this equation, we also get the following properties. 1.

Note

Note that the last condition does not hold in converse. Covariance only measures linear dependence, so if two random variables are nonlinearly dependent, they can still have zero covariance.

By a similar definition, two random variables are conditionally independent if and only if,

We can interpret this as saying given knowledge of , knowing more about doesn’t influence .

Inner Products §

If we have independent random variables and , imagine them as vectors in vector space, and observe that

looks like the Pythagorean theorem. If we define 🎳 Inner Product

then we get that the length of a random variable

Furthermore, the angle between and satisfies