{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

# 04rand - EECS 501 RANDOM VARIABLES Fall 2001 So far Used...

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

EECS 501 RANDOM VARIABLES Fall 2001 So far: Used sample space Ω and event space A to describe outcome. Now: Use a number to describe the outcome of an experiment. DEF: A random variable x is a mapping x : Ω → R (Ω=sample space). x associates a number with each outcome of an experiment. EX: Flip coin 100 times. x = (# heads ) 3 ; y =#heads in 1 st 10 flips. Note: Numbers represent outcomes, so sets of numbers represent events. Q: For what subsets F ⊂ R can we compute Pr [ x F ]? A1: Induced prob. space: Pr [ x F ] = Pr [ ω Ω : x ( ω ) F ] = Pr [ x - 1 ( F )]. Probability space (Ω , A , Pr ) and random variable x define ( preimage ) induced probability space ( R , A x , P x ) where P x [ F ] = Pr [ x - 1 ( F )]. EX: Suppose F ∈ A x P x [ F ] = 0 or 1. Prove x=constant with prob. 1. Huh? Heuristically, each sample point ω of Ω maps to same constant c . Proof: 1 = Pr [Ω] = Pr [ x - 1 ( R )] = Pr [ x - 1 ( n = -∞ [ n, n + 1))] = n = -∞ Pr [ x - 1 ([ n, n + 1))] = n = -∞ P x [[ n, n + 1)]. But: Given that P x [[ n, n + 1)] = 0 or 1. So P x [[ N, N + 1)] = 1 for some N .

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.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern