Slide 1

1Concentration of measure inequalities put an upper bound on the value of a random variable to deviate from its expectation.2Improbability

Markov inequality states that a non-negative function can reach a value k with probability at most the ratio between its expectation and k.Proof by story: if I have to give away some allotted amount money but care only about those who get above some threshold, to maximize that probability pick a promising set of people and give them that amount exactly, while all the rest get 0.

Chebyshev inequality limits how much a random variable can deviate from expectation as a function of its variance.Proof by story: imposed with some variance on the amount of oney given to individuals, to maximize probability of deviating by k, Ill choose my favorites, wholl get exactly k above the expectation, my least favorites wholl get exactly k less than the expectation, while all the rest get exactly the expectation. The fraction of favories plus least favories is the variance devided by k^2.

Chesnoff bound looks at the sum of independent variables.Now Im enforced to split the money by randomly and independently giving each dollar to a random person.3

Chernoff

The proof relies on Markov inequality, plus two tricks and simple algebra.

The first trick is exponentiation:The event of the sum larger than whatever is the same as the exponent of the sum larger than the exponent of whatever.Exponent of sum is the product of exponent of individual elements.Now the next trick is to take both sides to the tth power, whre t is going to be set at the end; t should be a positive small number close to 0.Now apply Markov inequality.Last piece of algebra: since these events are independent, the expectation of product is the product of expectation.

The outcome is a ratio. Now one tries to set t so as to optimize this ratio:

4Normal

http://mathworld.wolfram.com/GaltonBoard.html

5Chernoff