3225_hw1

# 3225_hw1 - Homework 1 Math 3225 1 Recall that events A 1 A...

This preview shows pages 1–2. Sign up to view the full content.

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

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Homework 1, Math 3225 September 10, 2010 1. Recall that events A 1 , ..., A k are independent if for every non-empty subset S of { 1 , ..., k } we have P ( ∩ s ∈ S A s ) = productdisplay s ∈ S P ( A s ) . (1) As a consequence of this, it turns out that this implies, and is equivalent to, the statement Claim. For i = 1 , 2 ..., k we have that for any set B in the σ-algebra generated by all the sets A j , j negationslash = i , P ( A i , B ) = P ( A i ) P ( B ) . In the special case i = 1 this would be saying that for any B ∈ σ ( A 2 , ..., A k ) – i.e. B is any set gotten by doing any number of in- tersections, unions and complements (which will turn out to be finite in number, since k < ∞ ) of the events A 2 , ..., A k – we must have that P ( A 1 , B ) = P ( A 1 ) P ( B ). This equivalence is somewhat tricky to prove; and to give you a taste of what is involved, your first problem will be to prove the following: Suppose that A, B, C, and D are independent events (using the (1)...
View Full Document

## This note was uploaded on 10/23/2011 for the course MATH 3225 taught by Professor Staff during the Spring '08 term at Georgia Tech.

### Page1 / 2

3225_hw1 - Homework 1 Math 3225 1 Recall that events A 1 A...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online