Chebyshev.

• Remind variance, standard deviation. $\sigma_{X}^{2}$= E[(X - $\mu_{X}^{}$)²]
• E[XY] = E[X]E[Y] if independent
• variance of independent variables: sum of variances
• Pr[| X - $\mu$|$\ge$t$\sigma$] = Pr[(X - $\mu$)²$\ge$t²$\sigma^{2}_{}$]$\le$1/t²
• binomial distribution. variance np(1 - p). stdev $\sqrt{n}$.
• requires (only) a mean and variance. less applicable but more powerful than markov
• Balls in bins: err /1 ln²n.

Two point sampling.

• pseudorandom generators. Motivation. Idea of randomness as (complexity theoretic) resource like space or time.
• pairwise independent vars.
• generating over Z_p.
• pairwise sufficient for chebyshev.
• Suppose RP algorithm using n bits.
• What do with 2n bits?
• two direct draws: error prob. 1/4.
• pseudorandom generators gives error prob. 1/t for t trials.
• $\mu$= t/2. $\sigma$= $\sqrt{t}$/2.
• error if no cert, i.e. Y - E[Y]$\ge$t/2, prob. 1/t.

Chernoff Bound

Intro

• Markov: Pr[f (X) > z] < E[f (X)]/z.
• Chebyshev used X² in f
• other functions yield other bounds
• Chernoff most popular

Theorem:

• Let X_i poisson (ie independent 0/1) trials, E[$\sum$X_i] = $\mu$

Pr[X > (1 + $$\delta$$)$$\mu$$] < $$\left[\vphantom{ \frac{e^\delta}{(1+\delta)^{(1+\delta)}}}\right.$$$${\frac{e^\delta}{(1+\delta)^{(1+\delta)}}}$$$$\left.\vphantom{ \frac{e^\delta}{(1+\delta)^{(1+\delta)}}}\right]^{\mu}_{}$$.

• note independent of n, exponential in $\mu$.

Proof.

• For any t > 0,

$$\delta \mu \delta \mu$$

$${\frac{E[\exp(tX)]}{\exp(t(1+\delta)\mu)}}$$

• 
  
• Use independence.

$$\prod$$

$$\le$$

• 
  
• $\prod$exp(p_i(e^t - 1)) = exp($\mu$(e^t - 1))

¬…       So overall bound is

$${\frac{\exp((e^t-1)\mu)}{\exp(t(1+\delta)\mu)}}$$

True for any t. Plug in t = ln(1 + $\delta$).

¬…       This in turn less than e^{- $\mu$$\delta^{2}$/4} for $\delta$< 2e - 1. (Less than 2^{- (1 + $\delta$)$\mu$} for larger $\delta$).

¬…       By same argument on exp(- tX),

Pr[X < (1 - $$\delta$$)$$\mu$$] < $$\left[\vphantom{ \frac{e^{-\delta}}{(1-\delta)^{(1-\delta)}}}\right.$$$${\frac{e^{-\delta}}{(1-\delta)^{(1-\delta)}}}$$$$\left.\vphantom{ \frac{e^{-\delta}}{(1-\delta)^{(1-\delta)}}}\right]^{\mu}_{}$$

bound e^{- $\delta^{2}$/2}.

Summary, Probability of deviation by relative error $\delta$< 1 is at most e^{- $\delta^{2}$$\mu$/3} in each direction. Large $\delta$gives 2^{- (1 + $\delta$)$\mu$}

• Trails off when $\delta$$\approx$$\sqrt{\mu}$, meaning absolute error is ``expected'' to be $\sqrt{\mu}$
• (note variance is less than $\mu$. Compare Chebyshev).
• If $\mu$= $\Omega$(log n), probability of constant deviation is O(1/n), Useful if polynomial number of events.

Remark: bound applies to any vars distributed in range [0, 1].

Basic applications:

• cn log n balls in c bins. max matches average (unlike n balls in n bins).
• Set balancing (book p. 73). minimize max bias. get 4$\sqrt{n\ln n}$.

Zillions of Chernoff applications; will see next time.

• Summary, Probability of deviation by relative error $\delta$< 1 is at most e^{- $\delta^{2}$$\mu$/3} in each direction. Large $\delta$gives 2^{- (1 + $\delta$)$\mu$}
• Trails off when $\delta$$\approx$$\sqrt{\mu}$, meaning absolute error is ``expected'' to be $\sqrt{\mu}$
• (note variance is less than $\mu$. Compare Chebyshev).
• If $\mu$= $\Omega$(log n), probability of constant deviation is O(1/n), Useful if polynomial number of events.

¬…       Basic applications:

o      Set balancing. minimize max bias. 4$\sqrt{n\ln n}$.

o      cn log n balls in c bins. max matches average.