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Chebychev , Hoffding , Chernoff

Chebychev , Hoffding , Chernoff. Histo 1. X = 2*Bin(300,1/2) – 300 E[X] = 0. Histo 2. Y = 2*Bin(30,1/2) – 30 E[Y] = 0. Histo 3. Z = 4*Bin(10,1/4) – 10 E[Z] = 0. Histo 4. W = 0 E[W] = 0. A natural question:.

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Chebychev , Hoffding , Chernoff

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  1. Chebychev,Hoffding, Chernoff

  2. Histo 1 X = 2*Bin(300,1/2) – 300 E[X] = 0

  3. Histo 2 Y = 2*Bin(30,1/2) – 30 E[Y] = 0

  4. Histo 3 Z = 4*Bin(10,1/4) – 10 E[Z] = 0

  5. Histo 4 W = 0 E[W] = 0

  6. A natural question: • Is there a good parameter that allow to distinguish between these distributions? • Is there a way to measure the spread?

  7. Variance and Standard Deviation • The variance of X, denoted by Var(X) is the mean squared deviation of X from its expected value m = E(X): • Var(X) = E[(X-m)2]. • The standard deviation of X, denoted by SD(X) is the square root of the variance of X.

  8. Computational Formula for Variance Claim: Var(X) = E[(X-m)2]. Proof: E[ (X-m)2] = E[X2 – 2m X + m2] E[ (X-m)2] = E[X2] – 2m E[X] + m2 E[ (X-m)2] = E[X2] – 2m2+ m2 E[ (X-m)2] = E[X2] – E[X]2

  9. Properties of Variance and SD • Claim: Var(X) >= 0. Pf: Var(X) = å (x-m)2 P(X=x) >= 0 • Claim: Var(X) = 0 iff P[X=] = 1.

  10. Markov

  11. Chebyshev’s Inequality Theorem: X is random variable on sample space S, and P(X=r) it’s probability distribution. Then for any positive real number r: (proof in book) In words: the probability of finding a value of X farther away from the mean than r is smaller than the variance divided by r^2. r

  12. Example: What is the probability that with 100 Bernoulli trials we find more than 89 or less than 11 successes when the prob. of success is ½. X counts number of successes. EX=100 x ½ =50 V(X) = 100 x ½ x ½ = 25. P(|X-50|>=40)<=25/40^2 = 1/64

  13. Proof

  14. Popular form (same) • “k standard deviations”

  15. Proof (without Markov)

  16. Example

  17. Hoffding (1963) • Let be random variables that for

  18. , , Chernoff

  19. Example • What is the probability of getting 25 or fewer heads in 100 coin tosses?

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