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":

 

In short:

X = {0, 1}

 

Note: We could have chosen Heads=100 and Tails=150 if we wanted! It is our choice.

So:

  • We have an experiment (such as tossing a coin)
  • We give values to each event
  • The set of values is a Random Variable

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 1, 2, 3 or 4, 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.

die

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}

 

Probability

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

  • P(X = 1) = 1/6
  • P(X = 2) = 1/6
  • P(X = 3) = 1/6
  • P(X = 4) = 1/6
  • P(X = 5) = 1/6
  • P(X = 6) = 1/6

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

Example: Toss three 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 = "number
of Heads"
HHH   3
HHT   2
HTH   2
HTT   1
THH   2
THT   1
TTH   1
TTT   0

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:

  • P(X = 3) = 1/8
  • P(X = 2) = 3/8
  • P(X = 1) = 3/8
  • P(X = 0) = 1/8

 

die

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


2nd
Die
1 2 3 4 5 6
1 2 3 4 5 6 7
2 3 4 5 6 7 8
3 4 5 6 7 8 9
4 5 6 7 8 9 10
5 6 7 8 9 10 11
6 7 8 9 10 11 12

There are 6 × 6 = 36 of them, and the Sample Space = {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}

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

  • 2 occurs just once, so P(X = 2) = 1/36
  • 3 occurs twice, so P(X = 3) = 2/36 = 1/18
  • 4 occurs three times, so P(X = 4) = 3/36 = 1/12
  • 5 occurs four times, so P(X = 5) = 4/36 = 1/9
  • 6 occurs five times, so P(X = 6) = 5/36
  • 7 occurs six times, so P(X = 7) = 6/36 = 1/6
  • 8 occurs five times, so P(X = 8) = 5/36
  • 9 occurs four times, so P(X = 9) = 4/36 = 1/9
  • 10 occurs three times, so P(X = 10) = 3/36 = 1/12
  • 11 occurs twice, so P(X = 11) = 2/36 = 1/18
  • 12 occurs just once, so P(X = 12) = 1/36

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

Solving

We can also solve a Random Variable equation.

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

P(X = 4) = 1/12, and P(X = 10) = 1/12

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

Notice the different uses of X and x:

  • X is the Random Variable "The sum of the scores on the two dice".
  • x is a value that X can take.

Continuous

Random Variables can be either Discrete or Continuous:

  • Discrete Data can only take certain values (such as 1,2,3,4,5)
  • Continuous Data can take any value within a range (such as a person's height)

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.

 

Summary

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

The set of possible values is called the Sample Space.

A Random Variable is given a capital letter, such as X or Z.

Random Variables can be discrete or continuous.