Zaymiey

📐 Mathematics  ·  Probability  ·  JEE

If X₁,...,Xₙ are iid with mean μ and variance σ², the central limit theorem says X̄ has approx distribution:

Answer: N(μ, σ²/n).

  • A N(μ, σ²)
  • B N(μ, σ²/n)
  • C N(0,1)
  • D N(nμ, nσ²)

Correct answer: B. N(μ, σ²/n)

Explanation: CLT: X̄ = (X₁+…+Xₙ)/n. E[X̄] = μ, Var(X̄) = σ²/n. For large n, X̄ ~ N(μ, σ²/n) regardless of original distribution. Standardised: Z = (X̄−μ)/(σ/√n) ~ N(0,1). Typically n ≥ 30 suffices.

UABA and BA onlyB onlyOutside both circles: complement of (A union B)

Venn diagram of the universal set U with events A and B, showing the intersection (A and B), the parts unique to each event, and the complement region outside both.

Concept context

Chance, events, conditional probability, and Bayes theorem

Read the full Probability notes →