The Chernoff Bounds Focusing on Examples

1. The Chernoff BoundsFocusing on Examples presented by Nam-juKwak 17th February 2010

2. Table of Contents • Outline • Definitions • Basic Forms of the Bounds • Derivation • Example A • Example B

3. Outline • Chernoff Bounds: techniques for bounding the probability that a random variable deviates far from its expectation. • We will study several basic forms of the Chernoff Bounds, followed by simple examples where we can find how they work in reality.

4. Definitions • X1, … , Xn: independent Bernoulli trials • p1, … , pn: Pr[Xi=1]=pi, Pr[Xi=0]=1-pi, and 0<pi<1

5. Basic Forms of the Bounds • When and δ>0, • When and 0<δ≤1,

6. Derivation Markov’s inequality • For any positive real t,

7. Derivation (cont.) • Since the right part is minimized when t=ln(1+δ) (by differentiation),

8. Example A • A group of gamblers win each game they play with probability of 1/3. Assuming that the outcomes of the games are independent, derive an upper bound on the probability that they have a winning season in a season lasting n games.

9. Example A • Settings • pi=1/3 for 1≤i≤n • Xi: 1 if the gambler wins; otherwise, 0. • μ=E[Yn]=n/3 • Pr[Yn>n/2]<(?)

10. Example A • When δ=1/2,

11. Example B • A group of gamblers hire a new coach, and critics revise their estimates of the probability of their winning each game to 0.75. What is the probability that the gamblers suffer a losing season assuming the critics are right and the outcomes of their games are independent of one another?

12. Example B • Settings • pi=0.75 for 1≤i≤n • Xi: 1 if the gambler wins; otherwise, 0. • μ=E[Yn]=0.75n • Pr[Yn<n/2]<(?)

13. Example B • When δ=1/3,

14. Q&A • Ask questions, if any. • The contents are based on Randomized Algorithms, 1995, written by Rajeev Motowani and PrabhakarRaghavan