# Random variables and Probability Distribution Functions

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

 Random Varialbes  “https://www.mathsisfun.com/data/random-variables.html

 Operations Research S. D. Sharma