Random Variables

A lot of this material is inspired by:

1. Wiki article on Random Variables

2. Khan academy videos

3. MIT OCW course

Assume you toss a coin once, the coin can land either as heads or tails. The outcome is unknown before the throw.

We can denote the sample space as: $\mathrm{\Omega }=\{H,T\}$

Now, let us say that if the coin lands as Heads, we win 1$ and we win 0$ if the coin lands as Tails. We can associate a function $W$ with the experiment such that W(Heads) = 1 and W(Tails) = 0

Random Variable

A random variable is a function mapping the sample space (for example: $\mathrm{\Omega }=\{H,T\}$) of an experiment to a measurement space $E=\{0,1\}$. Random variables are represented using capital letters.

While we may have generally seen a random variable W associated with a coin toss taking the values 1 for Heads, 0 for Tails, we might have chosen any other values and still have a random variable. As an example

$$\begin{array}{r}Y(\omega )=\{\begin{array}{ll}100,& \text{ if }\omega =\text{ heads }\\ -20,& \text{ if }\omega =\text{ tails }\end{array}\end{array}$$

If our experiment is to roll two die (6-faced die) and note the sum of the numbers on the top side.

The sample space for this example is:

$$\begin{array}{r}\begin{array}{rl}S=\{& (1,1),(1,2),\dots ,(1,6),\\ & (2,1),(2,2),\dots ,(2,6),\\ & \cdots ,\cdots ,\cdots ,\cdots \\ & (6,1),(6,2),\dots ,(6,6)\}\end{array}\end{array}$$

Based on this sample space, we can see that our random variable $Z$ denoting the sum of numbers on the top side takes the values $\{2,3,4,5,6,7,8,9,10,11,12\}$

Discrete Random Variable

A discrete random variable can take discrete values. For example, W representing the money we win if we toss a coin randomly. Or, Y the top face of a dice when the dice is rolled. Or, $Z$ denoting the sum of numbers on the top side of two dies.

Let us now take a different example. Our experiment is to pick a person at random from a university and measure their weight. The sample space would be the list of all the people in the university and the random variable $T$ would be weight corresponding to that randomly chosen person. The weight of that person would be a continuous scale.

Continuous Random Variable

A continuous random variable takes continuous values.