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A Random Variable is a set of possible values from a random experiment.
Example: Tossing a coin: we could get Heads or Tails.
1 4 2 X 1 10 X
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} Cyberduck 6 4 6 – ftp and sftp browser free.
Iflicks 2 4 3 download free. 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} Charles 4 2 5 – java http proxy and monitor.
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 |
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} Cyberduck 6 4 6 – ftp and sftp browser free.
Iflicks 2 4 3 download free. 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} Charles 4 2 5 – java http proxy and monitor.
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
X 1 X 4 X 3 0
You can also learn how to find the Mean, Variance and Standard Deviation of Random Variables.
Type To 1 5 X 4 5 Picture Frame
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.