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In probability theory, a Chernoff bound is an exponentially decreasing upper bound on the tail of a random variable based on its moment generating function. The minimum of all such exponential bounds forms the Chernoff or Chernoff-Cramér bound , which may decay faster than exponential (e.g. sub-Gaussian ).
The classical Chernoff bounds concern the sum of independent, nonnegative, and uniformly bounded random variables. In the matrix setting, the analogous theorem concerns a sum of positive-semidefinite random matrices subjected to a uniform eigenvalue bound.
The Chernoff bound of the Q-function is () ... As in the one dimensional case, there is no simple analytical formula for the Q-function. Nevertheless, ...
As a result of its generality it may not (and usually does not) provide as sharp a bound as alternative methods that can be used if the distribution of the random variable is known. To improve the sharpness of the bounds provided by Chebyshev's inequality a number of methods have been developed; for a review see eg. [12] [37]
The formula can be understood as follows: ... A sharper bound can be obtained from the Chernoff bound: ... Binomial distribution formula calculator;
In probability theory and statistics, the moment-generating function of a real-valued random variable is an alternative specification of its probability distribution.Thus, it provides the basis of an alternative route to analytical results compared with working directly with probability density functions or cumulative distribution functions.
Chernoff bound [ edit ] The probability that a Poisson binomial distribution gets large, can be bounded using its moment generating function as follows (valid when s ≥ μ {\displaystyle s\geq \mu } and for any t > 0 {\displaystyle t>0} ):
In statistics, the Bhattacharyya distance is a quantity which represents a notion of similarity between two probability distributions. [1] It is closely related to the Bhattacharyya coefficient, which is a measure of the amount of overlap between two statistical samples or populations.