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Markov's inequality (and other similar inequalities) relate probabilities to expectations, and provide (frequently loose but still useful) bounds for the cumulative distribution function of a random variable. Markov's inequality can also be used to upper bound the expectation of a non-negative random variable in terms of its distribution function.
In mathematics, the Markov brothers' inequality is an inequality, proved in the 1890s by brothers Andrey Markov and Vladimir Markov, two Russian mathematicians. This inequality bounds the maximum of the derivatives of a polynomial on an interval in terms of the maximum of the polynomial. [ 1 ]
Feller derives the equations under slightly different conditions, starting with the concept of purely discontinuous Markov process and then formulating them for more general state spaces. [5] Feller proves the existence of solutions of probabilistic character to the Kolmogorov forward equations and Kolmogorov backward equations under natural ...
In mathematics, the second moment method is a technique used in probability theory and analysis to show that a random variable has positive probability of being positive. More generally, the "moment method" consists of bounding the probability that a random variable fluctuates far from its mean, by using its moments.
The data processing inequality is an information theoretic concept that states that the information content of a signal cannot be increased via a local physical operation. This can be expressed concisely as 'post-processing cannot increase information'.
A simple proof by martingales is in the next section. Donald J. Newman and Lawrence Shepp gave a generalization of the coupon collector's problem when m copies of each coupon need to be collected. Let T m be the first time m copies of each coupon are collected.
Consider this figure depicting a section of a Markov chain with states i, j, k and l and the corresponding transition probabilities. Here Kolmogorov's criterion implies that the product of probabilities when traversing through any closed loop must be equal, so the product around the loop i to j to l to k returning to i must be equal to the loop the other way round,
Any function belongs to L 1,w and in addition one has the inequality ‖ ‖, ‖ ‖. This is nothing but Markov's inequality (aka Chebyshev's Inequality). The converse is not true. For example, the function 1/x belongs to L 1,w but not to L 1.