Expectation and variance
Objective. Calculate and interpret long-run measures.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: the key idea
Expectation E(X) is the long-run average value of a random variable, calculated as Σx·P(X=x), while variance Var(X) measures the spread of outcomes around that average using Var(X)=E(X²)−[E(X)]². For example, if X takes values 1 and 2 with probabilities 0.5 each, E(X)=1(0.5)+2(0.5)=1.5. These measures let you compare distributions and predict average behaviour over many repeated trials, and variance is always non-negative since it comes from a sum of squared deviations.
Worked example
X has distribution P(X=0)=0.4, P(X=1)=0.6. Find the expectation E(X).
- 1Use E(X)=Σx·P(X=x)
- 2E(X)=0(0.4)+1(0.6)
- 3E(X)=0+0.6
- 4E(X)=0.6
Answer: 0.6
Concept mastery
Based on your past attempts at this lesson — the question types to practise again come first.
Complete a practice set to see your concept breakdown here.
Practice
Practice happens on its own screen, one question at a time. Answers stay hidden until you submit yours, and you can stop and pick up at the same question later.
Start practisingCommon mistake. Students compute Var(X) as [E(X²)]−E(X) instead of E(X²)−[E(X)]², forgetting to square the mean. Always square E(X) before subtracting it from E(X²).
Your progress
Tick each step as you finish it. Your exact reading position is saved automatically, so Continue drops you back on the same line.
0 of 5 steps complete
Your place saves automatically as you read.