Computer and Communication Networks (paperback)

C.3. Multiple Random Variables

We often encounter several random variables that are related somehow. For example, a random signal as noise enters several circuits, and the outputs of these circuits can form multiple random variables . Multiple random variables are denoted by a vector X ={ X ₁ , X ₂ , ... X _n }.

C.3.1. Basic Functions of Two Random Variables

For two random variables X and Y , the joint cumulative distribution function denoted by F _X,Y ( x , y ), the joint probability mass function denoted by P _X,Y ( x , y ), and the joint probability density function , f _{X, Y} ( x , y ) are, respectively, derived from:

Equation C.25

Equation C.26

and

Equation C.27

We can define the marginal CDF of the two random variables as

Similarly, the marginal PMF of the two discrete random variables is

and the marginal PDF of the two continuous random variables is

C.3.2. Two Independent Random Variables

Two random variables are considered independent of each other if one of the following corresponding conditions is met:

Equation C.28

Equation C.29

or

Equation C.30