Expected Value of a Discrete Random Variable
Learn expected value for NSW Year 12 Mathematics Advanced (NESA outcome MAV-12-07). For a discrete random variable, the expected value is its long-run average — the mean result you would get over many repeats, found by weighting each value by its probability.
You will learn to calculate \(E(X)\) by multiplying each value by its probability and adding, find a missing probability from a known mean, and judge whether a game or bet is fair — the reasoning behind insurance prices and long-run decisions.
Theory
The expected value \(E(X)\) is the long-run average, weighting each value by its probability. This Year 12 Mathematics Advanced topic (MAV-12-07) computes \(E(X)\).
\(E(X)=\mu=\sum x\,P(x)\): multiply each value by its probability and add. It is the mean, a measure of centre, and need not be an attainable value (a die averages \(3.5\)).
Method
- Multiply each value by its probability.
- Add the products.
- Find a missing probability first if needed.
| \(E(X)\) | \(=\) | \(0.3+0.4=0.7\) |
| \(E(X)\) | \(=\) | \(\dfrac{21}{6}=3.5\) |
| \(E(X)\) | \(=\) | \(5(0.2)=\$1\) |
| \(E(X)\) | \(=\) | \(0.2+0.6+1.2=2\) |
Common pitfalls
Frequently asked questions
What is expected value?
It is the long-run average of a random variable, found by adding each value times its probability.
How do you calculate E(X)?
Multiply each value by its probability and add the results.
Can the expected value be a value X cannot take?
Yes. A fair die has expected value 3.5, which is never rolled.
What does E(X) represent?
The mean of the distribution, a measure of its centre.