This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Answers to Homework 6 AMS 570 4.2.1 Since { a n } and a are all constant random variables, P (  a n a  < ) can only be 0 or 1. It is equal to 1 when  a n a  < and 0 when  a n a  ≥ . a n → a, as n → ∞ ⇐⇒ ∀ > , ∃ N ∈ N such that P (  a n a  < ) = 1 , when n > N ⇐⇒ ∀ < ε < 1 , > , ∃ N ∈ N s.t. P (  a n a  < ) > 1 ε > , when n > N ⇐⇒ ∀ < ε < 1 , > , ∃ N ∈ N s.t. P (  a n a  < ) 1 < ε , when n > N ⇐⇒ ∀ > , lim n →∞ P (  a n a  < ) = 1 ⇐⇒ a n P→ a, as n → ∞ 4.2.2 (a) Y n ∼ b ( n,p ). Y n can be expressed as the sum of n independent Bernoulli(p) variables X 1 ,X 2 ,...,X n . That is Y n = n X i =1 X i ; P ( X i = 1) = p = 1 P ( X i = 0) , E( X i ) = p ∀ i. By the weak law of large numbers, X n = Y n n P→ E( X i ) = p. (b) g 1 ( x ) = 1 x is continuous at p, < p < 1. By Theorem 4.2.4, g 1 Y n n = 1 Y n n P→ 1 p....
View
Full
Document
This note was uploaded on 03/24/2010 for the course AMS 54039 taught by Professor Hongshikahn during the Spring '10 term at SUNY Stony Brook.
 Spring '10
 HongshikAhn

Click to edit the document details