8.2 CHEBYSHEV'S INEQUALITY AND THE WEAK LAW 8.3 The Central Limit Theorem Theorem3.1 The Central Limit Theorem Let X1,...,X be a sequence of independent and identically distributed random variables.Each having finite mean E(Xi)=u and variance D(Xi)=o2.Then the distribution of X+…+Xn-n4 aVn tends to the standard normal as noo.That is,for -00<a<十0， imP2X-业≤X刘=() n-0o Vna Xiaohan Yang Chapter 8 Limit Theorems logo 8.2 CHEBYSHEV’S INEQUALITY AND THE WEAK LAW OF LARGE NUMBERS 8.3 The Central Limit Theorem Theorem3.1 The Central Limit Theorem Let X1, · · · , Xn be a sequence of independent and identically distributed random variables. Each having finite mean E(Xi) = µ and variance D(Xi) = σ 2 . Then the distribution of X1 + · · · + Xn − nµ σ √ n tends to the standard normal as n → ∞. That is, for −∞ < a < +∞ß lim n→∞ P( Pn i=1 Xi − nµ √ nσ ≤ x) = Φ(x) Xiaohan Yang Chapter 8 Limit Theorems