The Chebyshev inequality is used to prove the Weak Law of Large Numbers.

In probability theory, Chebyshev ’ s inequality ( also spelled as Tchebysheff ’ s inequality ) guarantees that in any probability distribution ," nearly all " values are close to the mean — the precise statement being that no more than 1 / k < sup > 2 </ sup > of the distribution ’ s values can be more than k standard deviations away from the mean.

The term Chebyshev ’ s inequality may also refer to the Markov's inequality, especially in the context of analysis.

We can then infer that the probability that it has between 600 and 1400 words ( i. e. within k = 2 SDs of the mean ) must be more than 75 %, because there is less than chance to be outside that range, by Chebyshev ’ s inequality.

However, the bounds provided by Chebyshev ’ s inequality cannot, in general ( remaining sound for variables of arbitrary distribution ), be improved upon.

It is named after the Russian mathematician Andrey Markov, although it appeared earlier in the work of Pafnuty Chebyshev ( Markov's teacher ), and many sources, especially in analysis, refer to it as Chebyshev's inequality or Bienaymé's inequality.

It is a sharper bound than the known first or second moment based tail bounds such as Markov's inequality or Chebyshev inequality, which only yield power-law bounds on tail decay.

In mathematics, Chebyshev's sum inequality, named after Pafnuty Chebyshev, states that if

A second is the Vysochanskiï – Petunin inequality, a refinement of the Chebyshev inequality.

The Chebyshev inequality guarantees that in any probability distribution, " nearly all " the values are " close to " the mean value.

The Bertrand – Chebyshev theorem ( 1845 | 1850 ) states that for any, there exists a prime number such that < math > n < p < 2n </ math >.

It is also known as chessboard distance, since in the game of chess the minimum number of moves needed by a king to go from one square on a chessboard to another equals the Chebyshev distance between the centers of the squares, if the squares have side length one, as represented in 2-D spatial coordinates with axes aligned to the edges of the board.

On the other hand, the Lebesgue constant grows only logarithmically if Chebyshev nodes are used, since we have

The most common such generalized counting function is the Chebyshev function, defined by

For instance, in the case of the Chebyshev filter it is usual to define the cutoff frequency as the point after the last peak in the frequency response at which the level has fallen to the design value of the passband ripple.

That, in turn, is the source of the Russian name Пафнутий ( Pafnuty ), e. g. the famous mathematician Pafnuty Chebyshev.

* Chebyshev distance measures distance assuming only the most significant dimension is relevant.

* Minkowski distance is a generalization that unifies Euclidean distance, Manhattan distance, and Chebyshev distance.

The right hand side of the formula for cos ( nx ) is in fact the value T < sub > n </ sub >( cos x ) of the Chebyshev polynomial T < sub > n </ sub > at cos x.

Chebyshev is known for his work in the field of probability, statistics, mechanics, and number theory.

This is a consequence of the Chebyshev inequalities for the number of prime numbers less than, which state that is of the order of.

Chebyshev is also known for the Chebyshev polynomials and the Chebyshev bias-the difference between the number of primes that are 3 ( modulo 4 ) and 1 ( modulo 4 ).

Chebyshev is considered top be a founding father of Russian mathematics.

One common method, especially on higher-end processors with floating-point units, is to combine a polynomial or rational approximation ( such as Chebyshev approximation, best uniform approximation, and Padé approximation, and typically for higher or variable precisions, Taylor and Laurent series ) with range reduction and a table lookup — they first look up the closest angle in a small table, and then use the polynomial to compute the correction.

The theorem is named after Russian mathematician Pafnuty Chebyshev, although it was first formulated by his friend and colleague Irénée-Jules Bienaymé.

The letter T is used because of the alternative transliterations of the name Chebyshev as Tchebycheff ( French ) or Tschebyschow ( German ).

This relationship is used in the Chebyshev spectral method of solving differential equations.

His conjecture was completely proved by Chebyshev ( 1821 – 1894 ) in 1850 and so the postulate is also called the Bertrand-Chebyshev theorem or Chebyshev's theorem.

Chebyshev's theorem is a name given to several theorems proven by Russian mathematician Pafnuty Chebyshev

After Bernoulli and Poisson published their efforts, other mathematicians also contributed to refinement of the law, including Chebyshev, Markov, Borel, Cantelli and Kolmogorov and Khinchin ( who finally provided a complete proof of the LLN for arbitrary random variables ).

#* Chebyshev distance-the maximum absolute difference in values for any variable

McClellan, Chebyshev Approximation for Nonrecursive Digital Filters with Linear Phase, IEEE Trans.

Not having any teaching obligations, this allowed Lyapunov to focus on his studies and in particular he was able to bring to a conclusion the work on the problem of Chebyshev with which he started his scientific career.

Chebyshev filters have the property that they minimize the error between the idealized and the actual filter characteristic over the range of the filter, but with ripples in the passband.

The frequency response of a fourth-order type I Chebyshev low-pass filter with

Log of the absolute value of the gain of an 8th order Chebyshev type I filter in complex frequency space ( s = σ + jω ) with ε = 0. 1 and.

Gain and group delay of a fifth-order type I Chebyshev filter with ε = 0. 5.

The gain and the group delay for a fifth-order type I Chebyshev filter with ε = 0. 5 are plotted in the graph on the left.

The frequency response of a fifth-order type II Chebyshev low-pass filter with

Log of the absolute value of the gain of an 8th order Chebyshev type II filter in complex frequency space ( s = σ + jω ) with ε = 0. 1 and.

Gain and group delay of a fifth-order type II Chebyshev filter with ε = 0. 1.

The gain and the group delay for a fifth-order type II Chebyshev filter with ε = 0. 1 are plotted in the graph on the left.

As with most analog filters, the Chebyshev may be converted to a digital ( discrete-time ) recursive form via the bilinear transform.

Here is an image showing the Chebyshev filters next to other common kind of filters obtained with the same number of coefficients ( all filters are fifth order ):

In mathematics, the classical orthogonal polynomials are the most widely used orthogonal polynomials, and consist of the Hermite polynomials, the Laguerre polynomials, the Jacobi polynomials together with their special cases the Gegenbauer polynomials, the Chebyshev polynomials, and the Legendre polynomials.

A circle of radius r for the Chebyshev distance ( L < sub >∞</ sub > metric ) on a plane is also a square with side length 2r parallel to the coordinate axes, so planar Chebyshev distance can be viewed as equivalent by rotation and scaling to planar taxicab distance.

In chess, the distance between squares on the chessboard for rooks is measured in Manhattan distance ; kings and queens use Chebyshev distance, and bishops use the Manhattan distance ( between squares of the same color ) on the chessboard rotated 45 degrees, i. e., with its diagonals as coordinate axes.

, the only OEIS reference to 69105 is, " Partial sums of Chebyshev sequence S ( n, 16 )," with 69105 corresponding to n

Compared with a Chebyshev Type I / Type II filter or an elliptic filter, the Butterworth filter has a slower roll-off, and thus will require a higher order to implement a particular stopband specification, but Butterworth filters have a more linear phase response in the pass-band than Chebyshev Type I / Type II and elliptic filters can achieve.

are the vectors of the dual basis with respect to the basis of Chebyshev polynomials ( defined as

The Chebyshev polynomials of the second kind are orthogonal polynomials with respect to the Wigner semicircle distribution.