Statistics - Random Variable (Random quantity|Aleatory variable|Stochastic variable)

Thomas Bayes


Random variable is also known as:

  • random quantity,
  • aleatory variable,
  • or stochastic variable

A random variable represents the result of a random process.

The random variable value is the summary of many outcomeS (original variable) of a random phenomenon that describes the result of a random process.


  • the number of heads after a certain number of coin flips (Flipping a coin several times is a random process)
  • the heights of different people. (The growth of a person is a random process)

Random variable:


Many random variables depend not only on a chance but also on time. They evolve in time while being random at each particular moment.

A random variable that depend on time is called a stochastic process.

Example of random variable

  • number of concurrent user
  • all queue type:
    • number of jobs in a queue,
    • number of truck in a waiting queue
  • internet traffic
  • system available capacity
  • stock price
  • air temperatur


Random variables can be:


ie taking any of a specified finite or countable list of values, endowed with a probability mass function characteristic of the random variable's probability distribution;

Person's height

In an experiment a person may be chosen at random, and one random variable may be the person's height.

Mathematically, the random variable is interpreted as a function which maps the person to the person's height.

Associated with the random variable is a probability distribution that allows the computation of the probability that the height is in any subset of possible values, such as the probability that:

  • the height is between 180 and 190 cm,
  • the height is either less than 150 or more than 200 cm.

The person's number of children

This is a discrete random variable with non-negative integer values

Coin toss

Coin Flipping


See Statistics - Continuous Variable ie taking any numerical value in an interval or collection of intervals, via a probability density function that is characteristic of the random variable's probability distribution; or a mixture of both types.

Spinner that can choose a horizontal direction

The values taken by the random variable are directions.

<MATH> X = \text{the angle spun} </MATH>

Possible Sample space

  • North, West, East, South, Southeast, etc.
  • Degrees clockwise from North (real numbers from the interval [0, 360])

The probability:

  • of choosing a number in [0, 180] is ​1⁄2
  • density is 1/360.

Visualization / Central Limit Distribution

When random variables (independent) (estimate of a random process) are added to a set their distribution tends toward a normal distribution (informally a “bell curve”) See Statistics - Central limit theorem (CLT)

Documentation / Reference

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