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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 proof of Hoeffding's inequality follows similarly to concentration inequalities like Chernoff bounds. [9] The main difference is the use of Hoeffding's Lemma : Suppose X is a real random variable such that X ∈ [ a , b ] {\displaystyle X\in \left[a,b\right]} almost surely .
Chernoff bound, a bound on the tail distribution of sums of independent random variables, named for Herman Chernoff but due to Herman Rubin. [20] Cobb–Douglas, a production function named after Paul H. Douglas and Charles W Cobb, developed earlier by Philip Wicksteed.
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, ...
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.
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.
In probability theory, Bernstein inequalities give bounds on the probability that the sum of random variables deviates from its mean. In the simplest case, let X 1, ..., X n be independent Bernoulli random variables taking values +1 and −1 with probability 1/2 (this distribution is also known as the Rademacher distribution), then for every positive ,
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.