Variable and Random Variable

Random variable is a variable that is governed by chance. We first get acquainted with the word “variable” when we are taught to solve a simple algebraic equation like X + 5 = 7………(1). Here X is always equal to 2. A “random variable” is random in nature, so it takes different values unlike the variable X discussed in the equation (1).

Discrete Case

One dimensional random variable X

A player rolls a die once.  X denotes the number appearing on the face of a die. Unlike X discussed in equation (1), this number changes in every roll. Hence the X here is called a random variable and X = 1, 2, 3, 4, 5, 6. It takes values according to a probability distribution.

Two dimensional random variables (X₁, X₂)

Let’s consider a pair of simultaneous equations.

2X₁+ 3X₂ = 5

X₁-3X₂=2

X₁ = 7/3 and X₂ = 1/9 is the solution of the pair of simultaneous equations. These are called variables and take only these values for this pair of simultaneous equations.

Where as if two players roll a die one after the other, then the number appearing on the faces of dice thrown by these two players us denoted by two dimensional random variable (X₁, X₂). Here X₁ denotes number thrown by player 1 and X₂ denotes number thrown by player 2.  X₁= 1, 2, 3, 4,5 6  & X₂= 1, 2, 3, 4, 5, 6.  These are random in nature as they take different values in every roll and these values are governed by a probability distribution.