Limit Theorems

Abstract

Inequalities Proposition 1 (Markov Inequality) If X is a nonnegative random variable with mean µ, then for any a > 0, P(X ≥ a) ≤ µ a. Proposition 2 (Chebyshev Inequality) If X is a random variable with mean µ and variance σ 2 , then for any a > 0, P(|X − µ| ≥ a) ≤ σ 2 a 2. Example 3 Suppose that it is known that the number of items produced in a factory during a week is a random variable with mean 50. (a) What can be said about the probability that this week's production will exceed 75? (b) If the variance of a weeks production is known to be 25, what can be said about the probability that this week's production will between 40 and 60? Limit Theorems Theorem 4 (Weak Law of Large Numbers) Let X 1 , X 2 ,. .. be a sequence of independent and identically distributed random variables with common mean µ. Then for any > 0, lim n→∞ P X 1 + · · · + X n n − µ ≥ = 0. Proof We will prove this theorem under the additional assumption that random variables has a finite variance σ 2. Theorem 5 (Strong Law of Large Numbers) Let X 1 , X 2 ,. .. be a sequence of independent and identically distributed random variables with common mean µ. Then, P lim n→∞ X 1 + · · · + X n n = µ = 1. Theorem 6 (Central Limit Theorem) Let X 1 , X 2 ,. .. be a sequence of independent and identically distributed random variables with common mean µ and common variance σ 2. Then the distribution of X 1 + · · · + X n − nµ σ √ n