## 26 Dec 2018

### Random Variables (Discrete and Continuous Random Variables), Sample space and Random Variables Examples

Before understanding the concept of random variables, you need to know what is a sample space in probability.

## What is Sample Space in probability (S)

The range of all possible outcomes of an experiment is known as the 'Sample Space' (s).

Now let's understand the definition of Random Variables-

Probability Density Function (PDF) - Definition, Basics and Properties of Probability Density Function (PDF) with Derivation and Proof

## Random Variable (Random Variable Definition)

A random variable is a function which can take on any value from the sample space and having range of some set of real numbers, is known as the random variable of the experiment.

## Types of Random Variables

Random variables can be classified as-
#Discrete Random Variables and
#Continuous Random Variables

Now we will understand the Discrete Random Variables with the help of an example-

These are the random variable which can take on only finite number of values in a finite observation interval. So we can say that to discrete random variable has distinct values that can be counted.
We Will understand this with the help of an example-

Cumulative Distribution Function (CDF) - Properties of CDF - CDF Definition, Basics - Continuous and Discrete CDF

## Example of Discrete Random Variable

Let's take an example (experiment) of tossing 3 coins at the same time (simultaneously).
Now for this experiment the sample space is
S= {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}
Here let's suppose that the number of tails is the random variable X
So X= {0    1    1   1   2   2   2  3}
= {x1  x2  x3 x4 x5 x6 x7 x8}

Now let's discuss other type of random variables i.e.  Continuous random variables.

## Continuous Random Variables

A random variable that takes on an infinite number of values is known as a continuous random variable. Many physical systems (experiments) can produce infinite number of outputs in a finite time of observation. In such cases we use continuous random variables to define outputs of such systems.

Joint Probability Density Function (Joint PDF) - Properties of Joint PDF with Derivation- Relation Between Probability and Joint PDF

## Examples of Continuous Random Variables

Example 1- A random variable that measures the time taken in completing a job, is continuous random variable, since there are infinite number of times (different times) to finish that job.

Example 2 - Noise voltage that is generated by an electronic amplifier has a continuous amplitude. Therefore sample space (S) and random variable (X) both are continuous.