Expectation
For a non-negative discrete random variable ,
where : a weighted average of the values of , weighted by their probabilities.
Let be a countable set and be a discrete random variable.
For , the expectation of is defined by
Alternatively, consider
Then
So the expectation of is the weighted average of the values taken by , with weights given by the probabilities of taking those values.
General definition
Suppose is discrete. We can define and as in Definition 3.8.
If at least one of and is finite, then we can define the expectation of by
Otherwise, is not defined.
Properties
-
If , then .
-
If and , then .
-
If , then and .
-
If and are random variables, then .
-
Let and be integrable random variables. Then
-
Suppose that are non-negative random variables. Then
Proof of countable additivity of expectation
With countable, the identity for non-negative random variables follows by interchanging the sums:
Non-negativity is what licenses the interchange: every partial sum of non-negative terms may be reordered freely.
Expectation of a function of a random variable
Let and consider defined by . Then is a random variable, with
Proof of the function formula
Let . Then , so
The last equality regroups terms by the value of rather than .
Tail sum formula
Suppose and takes integer values. Then
Proof of the tail sum formula
For any integer ,
Applying this to pointwise and using the countable additivity of expectation for non-negative summands,
Expectation of an indicator
For an event and ,
The expectation of an indicator is the probability of the underlying event; this bridge is what lets indicator arguments translate between probabilities and expectations.
Related
Stated in
- Definition 3.8 (Expectation for Non-Negative Discrete Random Variables)§3.1 Expectation
- Definition 3.11 (Expectation for General Discrete Random Variables)§3.1 Expectation
- Proposition 3.14 (Properties of Expectation)§3.1 Expectation
- Proposition 3.16§3.1 Expectation
- Proposition 3.17§3.1 Expectation