Laws of large numbers

Objective. Explain convergence of sample averages.

1. Learn
2. Worked example
3. Practice
4. Feedback
5. Continue

Learn: the key idea

Laws of large numbers describe how sample averages behave as the number of observations grows, stating that for independent identically distributed variables X1,...,Xn with mean μ, the sample mean X̄n converges to μ as n → ∞. The weak law states convergence in probability, P(|X̄n − μ| > ε) → 0 for any ε > 0, while the strong law gives almost sure convergence. For fair coin tosses coded as 0/1, the weak law guarantees the proportion of heads approaches 0.5 as the number of tosses increases, though individual runs can still fluctuate.

Worked example

State what the weak law of large numbers guarantees about the sample mean X̄n of n i.i.d. variables with mean μ = 7 as n → ∞.

1 $Recall the weak law concerns convergence in probability$
2 $Write the formal statement P(|X̄n - μ| > ε) \to 0$
3 $Substitute the given mean μ = 7 into the statement$
4 $Interpret the convergence of X̄n$

Answer: X̄n converges in probability to 7

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 practising

Common mistake. A common misconception is believing the law of large numbers implies past outcomes influence future ones, the so-called gambler's fallacy, expecting a run of tails to make heads more likely next. The law only concerns long-run averages, not individual future trials.

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.

My dashboard