The weak law of large numbers guarantees that as n → ∞, the sample mean X̄ₙ will:
AEventually equal the population mean μ exactly, with certainty
BMake the probability of differing from μ by any fixed positive amount shrink to zero
CConverge to μ with probability 1 along every sample path
DEqual μ for all n larger than some threshold determined by the variance
The WLLN says: for any ε > 0, P(|X̄ₙ − μ| > ε) → 0 as n → ∞. This is 'convergence in probability' — you can make the chance of being far from μ as small as you like by taking n large enough. It does NOT say X̄ₙ will equal μ exactly (that would require infinitely many observations), and it does NOT guarantee that every sample path converges to μ (that is the stronger claim of the strong law). Options A and D describe a certainty the weak law does not provide.
Question 2 Multiple Choice
Why does averaging n independent observations reduce the error in estimating the population mean?
AAveraging cancels all outliers by symmetry, so only typical values remain
BThe variance of the sample mean is σ²/n, which shrinks as n grows, concentrating the distribution around μ
CThe law of averages ensures extreme values become rarer in larger samples
DLarger samples include a greater fraction of the population, making the sample more representative
The variance of X̄ₙ = (X₁ + ⋯ + Xₙ)/n is Var(X̄ₙ) = σ²/n, which follows from independence and the scaling of variance. As n increases, σ²/n → 0, meaning the distribution of X̄ₙ becomes increasingly concentrated around μ. By Chebyshev's inequality, P(|X̄ₙ − μ| > ε) ≤ σ²/(nε²) → 0. This is the precise mechanism: random fluctuations cancel out via variance reduction. Option A ('the law of averages') is a common misconception suggesting that extreme values are 'due' to be compensated — but past observations don't influence future ones.
Question 3 True / False
The weak law of large numbers applies for any fixed margin of error ε > 0, no matter how small, as long as the sample size is large enough.
TTrue
FFalse
Answer: True
This is the quantifier structure of the WLLN: for ANY ε > 0, P(|X̄ₙ − μ| > ε) → 0. There is no lower bound on ε — even ε = 0.000001 works. The catch is that the required n may be very large for tiny ε (since the bound σ²/(nε²) shows n must grow as ε shrinks). But the guarantee holds for every positive ε. This universality is what makes the WLLN the foundational justification for using sample averages.
Question 4 True / False
The weak law of large numbers and the strong law of large numbers make the same guarantee — both say the sample mean converges to the population mean.
TTrue
FFalse
Answer: False
The two laws describe different modes of convergence. The weak law (convergence in probability) says that for any fixed ε, the probability of being far from μ vanishes — but it says nothing about the long-run behavior of any single sample path. The strong law (almost sure convergence) says that the event {X̄ₙ → μ} has probability 1 — the sample path of X̄ₙ actually settles at μ with probability 1. The strong law is strictly more powerful. The weak law does not imply the strong law.
Question 5 Short Answer
What does 'convergence in probability' mean for the weak law of large numbers, and how does it differ from saying 'the sample mean will eventually equal the population mean'?
Think about your answer, then reveal below.
Model answer: Convergence in probability means: for any tolerance ε > 0, however small, the probability that X̄ₙ differs from μ by more than ε can be made arbitrarily small by taking n large enough. It is a statement about probabilities approaching zero, not about sample paths reaching an exact value. Saying the sample mean 'will eventually equal μ' would mean the outcome is certain after enough observations — which is false. Even with a million observations, there is still a small (but now tiny) probability of being far from μ. The WLLN says that probability can be made as small as desired, not that it ever reaches zero.
The distinction matters for how we use the WLLN in practice. It tells us that large samples give reliable estimates (the chance of a large error is small) without promising exact equality. This is the right foundation for statistics: confidence increases with sample size, but certainty is never achieved from finite samples.