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Bennett's inequality, an upper bound on the probability that the sum of independent random variables deviates from its expected value by more than any specified amount; Bhatia–Davis inequality, an upper bound on the variance of any bounded probability distribution; Bernstein inequalities (probability theory) Boole's inequality; Borell–TIS ...
In mathematics, the Khintchine inequality, named after Aleksandr Khinchin and spelled in multiple ways in the Latin alphabet, is a theorem from probability, and is also frequently used in analysis. Consider N {\displaystyle N} complex numbers x 1 , … , x N ∈ C {\displaystyle x_{1},\dots ,x_{N}\in \mathbb {C} } , which can be pictured as ...
Probability generating functions are particularly useful for dealing with functions of independent random variables. For example: If , =,,, is a sequence of independent (and not necessarily identically distributed) random variables that take on natural-number values, and
Another example of a complex random variable is the uniform distribution over the filled unit circle, i.e. the set {| |}. This random variable is an example of a complex random variable for which the probability density function is defined. The density function is shown as the yellow disk and dark blue base in the following figure.
Anti-concentration inequalities, on the other hand, provide an upper bound on how much a random variable can concentrate, either on a specific value or range of values. A concrete example is that if you flip a fair coin n {\displaystyle n} times, the probability that any given number of heads appears will be less than 1 n {\displaystyle {\frac ...
within a certain time period, say one year, arising from a random number N of individual insurance claims, whose sizes are described by the random variables (X n) n∈. Under the above assumptions, Wald's equation can be used to calculate the expected total claim amount when information about the average claim number per year and the average ...
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 ,
Hoeffding's inequality was proven by Wassily Hoeffding in 1963. [1] Hoeffding's inequality is a special case of the Azuma–Hoeffding inequality and McDiarmid's inequality. It is similar to the Chernoff bound, but tends to be less sharp, in particular when the variance of the random variables is small. [2]