lec15 - with the following sequence of PMFs: Does converge?...

Info iconThis preview shows pages 1–10. Sign up to view the full content.

View Full Document Right Arrow Icon
LECTURE 15 • Readings: Sections 7.1-7.3 Lecture outline • Limit theorems: –Chebyshev inequality –Convergence in probability
Background image of page 1

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

View Full DocumentRight Arrow Icon
Motivation i.i.d., What happens as ? (sample mean) •Why bo the r ?
Background image of page 2
Chebyshev’s Inequality • Random variable :
Background image of page 3

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

View Full DocumentRight Arrow Icon
Deterministic Limits : Review • We have a: — Sequence : —Number : • We say that converges to , and write: • If (intuitively): eventually gets and stays (arbitrarily) close to ”. • If (rigorously): For every there exists , such that for all , we have:
Background image of page 4
Convergence “in probability” • We have a sequence of random variables: • We say that converges to a number : “ (Almost) all of the PMF/PDF of eventually gets concentrated (arbitrarily) close to ”. • If (intuitively): • If (rigorously): For every , we have:
Background image of page 5

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

View Full DocumentRight Arrow Icon
Example • Consider a sequence of random variables
Background image of page 6
Background image of page 7

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

View Full DocumentRight Arrow Icon
Background image of page 8
Background image of page 9

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

View Full DocumentRight Arrow Icon
Background image of page 10
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: with the following sequence of PMFs: Does converge? What is ? Convergence of the Sample Mean (finite mean and variance ) i.i.d., Mean: Variance: Chebyshev : Limit: The Pollsters Problem : fraction of population that do . person polled: : fraction of Yes in our sample. Suppose we want: Chebyshev: But we have : Thus: So, let (conservative). Die Experiment (1) Unfair die, with probability of face . Independent throws: Thus, are i.i.d. with PMF: Define: Let: frequency of face Die Experiment (2) is Bernoulli with probability , thus: Then: Chebyshev: It follows that: Therefore, the sample frequency of each face converges in probability to the probability of that face. This allows us to do simulations....
View Full Document

This note was uploaded on 12/08/2010 for the course MATH 6.041 / 6. taught by Professor Muntherdahleh during the Spring '10 term at MIT.

Page1 / 10

lec15 - with the following sequence of PMFs: Does converge?...

This preview shows document pages 1 - 10. Sign up to view the full document.

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