A rule that assigns a single real number to each outcome (sample point) of the random experiment is called random variable.

**Definition
of random variable**

**Definition**

Let S be the sample space of a random experiment. A rule that
assigns a single real number to each outcome (sample point) of the random
experiment is called random variable.

In other words, a random variable is a real valued function
defined on a sample space S that is with each outcome ω of a random experiment
there corresponds a unique real value x known as a value of the random variable
X. That is *X*(ω ) = *x*.

Generally random variables are denoted by upper case alphabets
like *X*, *Y*, *Z* … and their values
or realizations are denoted by the corresponding lower case letters. For
example, if *X* is a random variable,
the realizations are *x*_{1}, *x*_{2} …

Consider the random experiment of rolling a die.

The sample space of the experiment is *S*={1, 2, 3, 4, 5, 6}

Let *X* denotes the face
of the die appears on top.

The assigning rule is

*X*(1) = 1,* X*(2) = 2,* X*(3) = 3,* X*(4)=4,* X*(5)=5 and* X*(6)=6

Hence the values taken by the random variable *X* are 1,2,3,4,5,6. These values are also
called the realization of the random variable *X*.

Random experiment : Two coins are tossed simultaneously.

Sample space : *S*={*HH*, *HT*,
*TH*, *TT*}

Assigning rule : Let *X*
be a random variable defined as the number of heads comes up.

Here, the random variable *X*
takes the values 0, 1, 2 .

Experiment : Two dice are rolled simultaneously.

Sample space : {(1, 1),(1, 2),(1, 3),…(6, 6)}

Assigning rule : Let *X*
denote the sum of the numbers on the faces of dice

then *X _{ij }*=

Then *X* is a random
variable which takes the values 2, 3, 4 . .… 12.

That is the range of *X*
is {2, 3, 4…… 12}

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11th Statistics : Chapter 9 : Random Variables and Mathematical Expectation : Definition of random variable |

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