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What is a random variable?

Module by: Brandon Hodgson

What is a random variable?

A Random Variable is a function, which assigns unique numerical values to all possible outcomes of a random experiment under fixed conditions(Ali 2000). A random variable is not a variable but rather a function that maps events to numbers (Wikipedia 2006).

Example 1

This example is extracted from (Ali 2000). Suppose that a coin is tossed three times and the sequence of heads and tails is noted. The sample space for this experiment evaluates to: S={HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}. Now let the random variable X be the number of heads in three coin tosses. X assigns each outcome in S a number from the set Sx={0, 1, 2, 3}. The table below lists the eight outcomes of S and the corresponding values of X.

Outcome HHH HHT HTH THH HTT THT TTH TTT
X 3 2 2 2 1 1 1 0

X is then a random variable taking on values in the set SX = {0, 1, 2, 3}.

Mathematically, a random variable is defined as a measurable function from a probability space to some measurable space (Wikipedia 2006). This measurable space is the space of possible values of the variable, and it is usually taken to be the real numbers (Wikipedia 2006).

The condition for a function to be a random variable is that the random variable cannot be multivalued (Ali 2000).

There are three types of random variables:

  • A Continuous Random Variable is one that takes an infinite number of possible values (Ali 2000). Example: Duration of a call in a telephone exchange.
  • A Discrete Random Variable is one that takes a finite distinct values (Ali 2000). Example: A number of students who fail a test.
  • A Mixed Random Variable is one for which some of its values are continuous and some are discrete (Ali 2000).

Exercise 1

Can measurements of power (in dB) received from an antenna be considered a random variable?

Solution 1

Yes. Specifically it should be considered as a continuous random variable as the power of any signal attenuates through a transmission line. The attenuation factors associated with each transmission line are only approximate. Thus the power received from the antenna can take any value.

References:

Ali M. "Probabilistic Sytems Analyses: Part I," University of the Witwatersrand, 2000.

Wikipedia. "Random Variable," Wikimedia Foundation Inc, http://en.wikipedia.org/wiki/Random_variable, Last accessed 17 February 2006.

Author of assignment 3: Brandon Hodgson

Author of assignment 1: Mphekwane (Eddie) Mamahlodi

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