Chernoff bounds The Chernoff bound for a random
Chernoff bounds The Chernoff bound for a random variable X is obtained as follows: for any t >0, Pr[X a] = Pr[et. X eta] ≤ E[et. X ] / eta Similarly, for any t <0, Pr[X ≤ a] = Pr[et. X eta] ≤ E[et. X ] / eta The value of t that minimizes E[et. X ] / eta gives the best possible bounds.
Moment generating functions • Def: The moment generating function of a random variable X is MX(t) = E[et. X]. • E[Xn] = MX(n)(0) , which is the nth derivative of MX(t) evaluated at t = 0. • Fact: If MX(t)= MY(t) for all t in (-c, c) for some c > 0, then X and Y have the same distribution. • If X and Y are independent r. v. , then MX+Y(t)= MX(t) MY(t).
Chernoff bounds for the sum of Poisson trials • Poisson trials: the distribution of a sum of independent 0 -1 random variables, which may not be identical. • Bernoulli trials: same as above except that all the random variables are identical. • Xi: i=1…n, mutually independent 0 -1 r. v. with Pr[Xi=1]=pi. Let X =X 1+…+Xn and E[X] =μ=p 1+. . . +pn. MXi(t) =E[et. Xi] = piet +(1 -pi) = 1 + pi (et -1) ≤ exp (pi (et -1) ).
Chernoff bound for a sum of Poisson trials •
Proof •
Application: Estimating a parameter •
Better bound for special cases •
Better bound for special cases •
Better bound for special cases
Proof of set balancing •
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