# 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 choose Heads=100 and Tails=150 or other values if we want! 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 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}

## 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: 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 = "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

### 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 | |||||||
---|---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | ||

2nd Die |
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** 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:

- 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)**?

## Solving

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:

- 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.