Expected Value of a Discrete Random Variable
The expected value is the long-run average of a discrete random variable, \(E(X)=\sum x\,P(X=x)\).
Part of the NSW Year 12 Mathematics Advanced course, in the Statistical analysis area of study (Random variables focus area) of the 2024 syllabus. Work through practice questions with fully worked solutions and video lessons, or scroll down for the theory summary and worked examples.
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.