To understand random variable first you have to know about events. I will make you understand using examples.
What are events?
Example
Tossing three coins on which one head turns up
{HT,HH}
A random variable quantify events of occurence.
In other words, a random variable X is a function from set of events to real number.
Suppose S is set of events then X : S->R
Example-
If you toss four coins the possible out comes are
TTTT
TTTH
TTHT
TTHH
THTT
THTH
THHT
THHH
HTTT
HTTH
HTHT
HTHH
HHTT
HHTH
HHHT
HHHH
make a statement that two atleast one head turns up
X(TTTT)=0
X(TTTH)=1
X(TTHT)=1
X(TTHH)=2
X(THTT)=1
X(THTH)=2
X(THHT)=2
X(THHH)=3
X(HTTT)=1
X(HTTH)=2
X(HTHT)=2
X(HTHH)=3
X(HHTT)=2
X(HHTH)=3
X(HHHT)=3
X(HHHH)=4
Probability Distribution Function (PDF)-
“Different values of random variable togethor with their probabilities form probability distribution”
(Operations Research S. D. Sharma)
For Example
In other words, P is a Probability Distribution Function (PDF) defined as P : X(e)-> [0,1], where e is set of events.
For example
P(X(TTTT))=0
P(X(TTTH))= 1/4
P(X(TTHT))=1/14
P(X(TTHH))= 1/16
P(X(THTT))=1/4
P(X(THTH))= 1/16
P(X(THHT))=1/16
P(X(THHH))=1/64
Properties of the probability distribution function
1- P(xi)=>0
2 – Summation of all probabilities is 1 i.e ∑∞i=0 P(xi)=1
References
[1] Random Varialbes “https://www.mathsisfun.com/data/random-variables.html”
[2] Operations Research S. D. Sharma