A probability distribution can be summarized by its center and its spread.
Suppose an order produces either 0 or 10 units of extra value. If the larger outcome occurs 40% of the time, the long-run average is 4.
Expected value is not necessarily observed in a single trial. It is the average that emerges across repeated trials.
Key question
Where is the center of a distribution, and how widely do values spread around it?
What does expected value mean?
Expected value of a discrete variable
E(X) = Σ x·P(X=x)
| Outcome x | Probability | x × probability |
|---|---|---|
| 0 | 0.60 | 0.00 |
| 10 | 0.40 | 4.00 |
| Total | 1.00 | 4.00 |
The expectation need not be a possible outcome
The expected value of a fair die is 3.5 even though no roll can equal 3.5.
Why do we need variance?
Expected value alone does not show whether values cluster tightly or swing between extremes. Variance averages the squared distance from the expectation.
Expected value
The long-run average of the distribution
Variance
How far values spread around the expectation
Standard deviation
The square root of variance, in the original unit
Same expectation
A: always 5
B: 0 half the time and 10 half the time
Both have expected value 5, but B has much larger variance.
How are variance and standard deviation related?
Because variance squares distances, its unit is squared. Standard deviation is the square root of variance and is therefore easier to interpret on the original scale.
| Measure | Unit | Advantage |
|---|---|---|
| Variance | original unit² | convenient for mathematical work |
| Standard deviation | original unit | easier to interpret |
Equal averages can carry different risk
Two campaigns can have the same average return while one is stable and the other alternates between large gains and large losses.
Decision making
Expected value describes the average outcome; variance and standard deviation describe uncertainty around it.
Linearity
E(aX+b) = aE(X)+b. Expectations transform linearly.
Change the probability and outcome size
Change the probability and size of the high outcome and watch expectation and variance update.
Expected value
4.00
Variance
24.00
Standard deviation
4.90
Key lesson
Expected value summarizes the center; variance summarizes uncertainty around that center.