Events and are independent if

Intuitively, it means that if we know the probability of event , this does not tell us anything about event , and whether it has occurred.

In addition, if and and and are independent, then the Conditional probability of as well as .

Random variables

Two Random variables are independent if

for all . In terms of the CDF and joint CDF, we can also write


If and are continuous with joint PDF , then the following statements are all equivalent:

  • and are independent.
  • for all . That is, we can factor the joint PDF to be the product of two functions and .
  • for all such that .
  • for all such that .

Expectation of their product

If and are independent r.v.s, then . Note that it is not always true the other way around (the equation being true doesn’t imply independence).


We first know that is true. Then by 2D LOTUS, we know that . The inner integral can be refactored to become , and the integral is now simply the definition of the PDF, which integrates to 1.

The complete integral is now . In this context, is constant because it does not depend on , so we can take it outside the integral, leaving us the definition of the PDF again. Therefore, we get .