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 0, 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 represents the Random Variable “The sum of the scores on the two dice”.
  • x represents 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.

 

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.

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