Independent Random Variables

Definition

Two random variables and are independent if, for every and ,

and we write . Otherwise, we say that and are dependent.

In principle, to check whether and are independent, we must use the above to check for all subsets and . Fortunately, we can use the following result.

Theorem

Let and have joint PDF . Then if and only if for all values and .

The following result is helpful for verifying independence.

Theorem

Suppose that the range of and is a (possibly infinite) rectangle. If for some functions and (not necessarily probability density functions), then and are independent.

Example

Let and have density

The range of and is the rectangle . We can write where and . Thus, .

Definition

If are independent and each has the same marginal distribution with CDF , we say that are IID (independent and identically distributed) and we write

If has a density , we also write . We also call a random sample of size from .

Sources

• Wasserman, L. (2010). All of Statistics: A concise Course in Statistical Inference. Chapter 2.7.