Lecture12 - Central Limit Theorem 11/08/2005 A More General...

Info iconThis preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon
Central Limit Theorem 11/08/2005
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
A More General Central Limit Theorem Theorem. Let X 1 , X 2 , . . . , X n , . . . be a sequence of indepen- dent discrete random variables, and let S n = X 1 + X 2 + ··· + X n . For each n , denote the mean and variance of X n by μ n and σ 2 n , respectively. Define the mean and variance of S n to be m n and s 2 n , respectively, and assume that s n → ∞ . If there exists a constant A , such that | X n | ≤ A for all n , then for a < b , lim n →∞ P ± a < S n - m n s n < b = 1 2 π Z b a e - x 2 / 2 dx . 1
Background image of page 2
Let S n be the number of successes in n Bernoulli trials with probability .8 for success on each trial. Let A n = S n /n be the average number of successes. In each case give the value for the limit, and give a reason for your answer. 1. lim n →∞ P ( A n = . 8) . 2. lim n →∞ P ( . 7 n < S n < . 9 n ) . 3. lim n →∞ P ( S n < . 8 n + . 8 n ) . 4.
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 4
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 07/16/2010 for the course MATH 20 taught by Professor Ionescu during the Fall '05 term at Dartmouth.

Page1 / 13

Lecture12 - Central Limit Theorem 11/08/2005 A More General...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online