A random variable, X is a deterministic function that assigns each possible outcome
from the sample space, S of an experiment to a real number.
A random variable is a function of the outcome of an experiment. Therefore, since each
outcome has a probability, the number assigned to that function by a random variable,
also has a probability. Two ways to analyse experiments via random variables is through
the Cumulative Distribution Function (CDF) and probability density function (pdf).
Cumulative Distribution Function
Let X be a random variable defined on a sample space S. Then Cumulative Distribution
Function of X, denoted by FX (x0 ), is defined as
FX (x0 ) = P (X ≤ x0 )
where, P (X ≤ x0 ) is the probability that value of the random variable X is less than or
equal to a real number x0 .
It has the following properties:
Property 1. 0 ≤ FX (x0 ) ≤ 1.
Property 2. FX (+∞) = 1, and FX (−∞) = 0.
Property 3. For x1 ≤ x2 , FX (x1 ) ≤ FX (x2 ).