# More on Random Variables

Expectation :: Properties :: Pairs :: Independence :: Two R.V.S. :: Functions :: Inequalities :: Conditional :: General

## Independence of r.v.s

- and are independent iff .
- if and are jointly continuous, then they are independent iff
- If and are discrete, then they are independent iff .
- If and are jointly Gaussian random variables, then they are independent iff . (Show this using the p.d.f.)

**Caution:**Gaussian r.v.s are special this way. As a general rule,

**uncorrelated does not imply independence.**.

In practice, it is common to assume that random variables are independent based on physical arguments, rather than to prove is by identifying a joint density and computing the marginals.

Many times, independence is also taken as an assumption, even when it is not strictly true. This independence assumption frequently simplifies analysis. However, the validity of the assumption must be validated (e.g., using computer simulations).