Concentration inequality
In mathematics, concentration inequalities provide probability bounds on how a random variable deviates from some value (e.g. its expectation). The laws of large numbers of classical probability theory state that sums of independent random variables are, under very mild conditions, close to their expectation with a large probability. Such sums are the most basic examples of random variables concentrated around their mean. Recent results shows that such behavior is shared by other functions of independent random variables.
Markov's inequality
If X is any random variable and a > 0, then
Proof can be found here.
We can extend Markov's inequality to a strictly increasing and non-negative function . We have
Chebyshev's inequality
Chebyshev's inequality is a special case of generalized Markov's inequality when
If X is any random variable and a > 0, then
Where Var(X) is the variance of X, defined as:
Asymptotic Behavior of Binomial Distribution
If a random variable X follows the binomial distribution with parameter and . The probability of getting exact successes in trials is given by the probability mass function
Let and 's are i.i.d. Bernoulli random variables with parameter . follows the binomial distribution with parameter with parameter and . Central Limit Theorem suggests when , is approximately normally distributed with mean and variance , and
For , where is a constant, the limit distribution of binomial distribution is the Poisson distribution
General Chernoff Inequality
Chernoff bound gives exponentially decreasing bounds on tail distributions of sums of independent random variables. Let denote i.i.d. random variables, satisfying , for . we have,
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