A random variable (r.v.) is a real valued function defined on a sample space S and taking values in (– ∞, ∞) or whose possible values are numerical outcomes of a random experiment.

**Definition
of a random variable**

A random variable
(r.v.) is a real valued function defined on a sample space *S* and taking
values in (– ∞, ∞) or whose possible values are numerical outcomes of a random
experiment.

**Note**

(i) If *x* is a
real number, the set of all ω in *S* such that *X*(ω)=*x* is,
denoted by *X* = *x* . Thus *P*(*X* = *x*) = *P*{ ω:*X*(ω)
= *x*}.

(ii) *P*(*X *__<__*
a*) =* P*{ ω:*X*(ω)* *∈(– ∞,* a*]} and*
P*(*a *<* X *__<__* b*) =* P*{ ω* *:* X*(ω)*
*∈(*a*,* b*]}.

(iii) One-dimensional
random variables will be denoted by capital letters, *X *,* Y *,* Z*,
...,* *etc. A typical outcome of the experiment will be denoted by* *w.* *Thusb *X*(ω)
represents the real number which the random variable *X* associates with
the outcome w. The values which *X*
, *Y* , *Z*, ..., etc, can assume are denoted by lower case letters, *viz*
., *x*, *y* , *z*, ..., etc.

Random variables are
classified into two types namely discrete and continuous random variables.
These are important for practical applications in the field of Mathematics and
Statistics. The above types of random variable are defined with examples as
follows.

Study Material, Lecturing Notes, Assignment, Reference, Wiki description explanation, brief detail

12th Business Maths and Statistics : Chapter 6 : Random Variable and Mathematical Expectation : Definition and Types of Random Variable |

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