Definition and Types of Random Variable

Definition of a random variable

Definition 6.1

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.

Types of Random Variable:

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.