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Bernoulli's inequality can be proved for case 2, in which is a non-negative integer and , using mathematical induction in the following form: we prove the inequality for r ∈ { 0 , 1 } {\displaystyle r\in \{0,1\}} ,
For instance, to solve the inequality 4x < 2x + 1 ≤ 3x + 2, it is not possible to isolate x in any one part of the inequality through addition or subtraction. Instead, the inequalities must be solved independently, yielding x < 1 / 2 and x ≥ −1 respectively, which can be combined into the final solution −1 ≤ x < 1 / 2 .
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
The bounds these inequalities give on a finite sample are less tight than those the Chebyshev inequality gives for a distribution. To illustrate this let the sample size N = 100 and let k = 3. Chebyshev's inequality states that at most approximately 11.11% of the distribution will lie at least three standard deviations away from the mean.
In mathematics, an inequation is a statement that an inequality holds between two values. [1] [2] It is usually written in the form of a pair of expressions denoting the values in question, with a relational sign between them indicating the specific inequality relation. Some examples of inequations are:
One method of solving elementary functional equations is substitution. [citation needed] Some solutions to functional equations have exploited surjectivity, injectivity, oddness, and evenness. [citation needed] Some functional equations have been solved with the use of ansatzes, mathematical induction. [citation needed]
Numerical methods for solving first-order IVPs often fall into one of two large categories: [5] linear multistep methods, or Runge–Kutta methods.A further division can be realized by dividing methods into those that are explicit and those that are implicit.
The exponential function is convex, so by Jensen's inequality (). It follows that the bound on the right tail is greater or equal to one when a ≤ E ( X ) {\displaystyle a\leq \operatorname {E} (X)} , and therefore trivial; similarly, the left bound is trivial for a ≥ E ( X ) {\displaystyle a\geq \operatorname {E} (X)} .
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