Random Variables

A Random Variable is a set of possible values from a random experiment.

Example: Tossing a coin: we could get Heads or Tails.

Let's give them the values Heads=0 and Tails=1 and we have a Random Variable "X":


random variable 1

In short:

X = {0, 1}

Note: We could choose Heads=100 and Tails=150 or other values if we want! It is our choice.


Not Like an Algebra Variable

In Algebra a variable, like x, is an unknown value:

Example: x + 2 = 6

In this case we can find that x=4

But a Random Variable is different ...

A Random Variable has a whole set of values ...

... and it could take on any of those values, randomly.


Example: X = {0, 1, 2, 3}

X could be 0, 1, 2, or 3 randomly.

And they might each have a different probability.

Capital Letters

We use a capital letter, like X or Y, to avoid confusion with the Algebra type of variable.

Sample Space

A Random Variable's set of values is the Sample Space.


Example: Throw a die once

Random Variable X = "The score shown on the top face".

X could be 1, 2, 3, 4, 5 or 6

So the Sample Space is {1, 2, 3, 4, 5, 6}


We can show the probability of any one value using this style:

P(X = value) = probability of that value

Example (continued): Throw a die once

X = {1, 2, 3, 4, 5, 6}

In this case they are all equally likely, so the probability of any one is 1/6

Note that the sum of the probabilities = 1, as it should be.

coin headcoin tailcoin head

Example: How many heads when we toss 3 coins?

X = "The number of Heads" is the Random Variable.

In this case, there could be 0 Heads (if all the coins land Tails up), 1 Head, 2 Heads or 3 Heads.

So the Sample Space = {0, 1, 2, 3}

But this time the outcomes are NOT all equally likely.

The three coins can land in eight possible ways:

  X =  
of Heads"
HHH coin headcoin headcoin head3
HHT coin headcoin headcoin tail2
HTH coin headcoin tailcoin head2
HTT coin headcoin tailcoin tail1
THH coin tailcoin headcoin head2
THT coin tailcoin headcoin tail1
TTH coin tailcoin tailcoin head1
TTT coin tailcoin tailcoin tail0

Looking at the table we see just 1 case of Three Heads, but 3 cases of Two Heads, 3 cases of One Head, and 1 case of Zero Heads. So:


2 dice

Example: Two dice are tossed.

The Random Variable is X = "The sum of the scores on the two dice".

Let's make a table of all possible values:

1st Die


There are 6 × 6 = 36 possible outcomes, and the Sample Space (which is the sum of the scores on the two dice) is {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}

Let's count how often each value occurs, and work out the probabilities:

A Range of Values

We could also calculate the probability that a Random Variable takes on a range of values.

Example (continued) What is the probability that the sum of the scores is 5, 6, 7 or 8?

In other words: What is P(5 ≤ X ≤ 8)?

P(5 ≤ X ≤ 8) =P(X=5) + P(X=6) + P(X=7) + P(X=8)
= (4+5+6+5)/36
= 20/36
= 5/9


We can also solve a Random Variable equation.

Example (continued) If P(X=x) = 1/12, what is the value of x?

Looking through the list above we find:

So there are two solutions: x = 4 or x = 10

Notice the different uses of X and x:


Random Variables can be either Discrete or Continuous:

All our examples have been Discrete.

Learn more at Continuous Random Variables.

Mean, Variance, Standard Deviation

You can also learn how to find the Mean, Variance and Standard Deviation of Random Variables.