Random Variables
A random variable turns outcomes into numbers.
If we roll a die, one random variable might be:
X = the number shown on the die
Another random variable might be:
Y = 1 if the roll is even
Y = 0 otherwise
The word "variable" can be misleading at first. A random variable is a function from outcomes to values. The randomness comes from the outcome we get.
For the die example, the outcome is the roll. The random variable Y maps each
roll to either 1 or 0:
1 -> 0
2 -> 1
3 -> 0
4 -> 1
5 -> 0
6 -> 1
So the event "roll is even" and the random variable Y are related but not the
same object. The event is the set {2, 4, 6}. The random variable is the rule
that returns 1 on that event and 0 otherwise.
Why Random Variables Help
Once outcomes become numbers, we can compute with them.
We can ask for:
- expected value
- variance
- probability of a range
- relationships between variables
ML Reading
In ML, labels, inputs, predictions, and losses can all be treated as random variables.
For example, a label Y may be random because different examples have different
labels. A prediction may be random because it depends on sampled data, model
parameters, or stochastic generation.
Loss is also often treated as a random variable. Each sampled example may produce a different loss value. Training tries to reduce average loss over the data-generating process.
For a die roll, define Y = 1 if the roll is even and 0 otherwise.
What is Y when the roll is 4?
Compute it first, then check your number.
Hint
Check whether 4 is even.
Solution
4 is even, so Y = 1. The random variable maps outcomes in the even event to
the number 1.
For the same random variable Y, what is Y when the roll is 5?
Compute it first, then check your number.
Hint
Check whether 5 is even.
Solution
5 is not even, so Y = 0. Outcomes outside the even event are mapped to the
number 0.
Is Y = 1 if the roll is even, 0 otherwise a random variable or an event?
Answer it first, then check.
Hint
It assigns a number to each outcome.
Solution
It is a random variable because it maps each outcome to a number.
If each sampled example produces a different loss value, can the loss be treated as a random variable?
Answer it first, then check.
Hint
A random variable is a numerical quantity depending on uncertain outcomes.
Solution
Yes. If the sampled example is uncertain and the loss is a number computed from that example, the loss can be treated as a random variable.
For a die roll, let Y = 1 if the roll is even and 0 otherwise.
Enter 1 if the event is {2, 4, 6} and Y is the numerical rule attached to
that event.
Compute it first, then check your number.
Hint
An event is a subset of the sample space. A random variable returns numbers.
Solution
Enter 1. {2,4,6} is the event. Y is the function that maps rolls in that
event to 1 and all other rolls to 0.
Before Moving On
A random variable lets us use arithmetic on uncertain outcomes.