The goal of Chebyshev's in-equality is to bound the probability that the RV is far from its mean (in either direction). This generally gives a stronger bound than Markov's inequality; if we know the variance of …

The Chebyshev polynomials denoted Tn(x) for n = 0, 1, . . . are a set of orthogonal polynomials on the open interval (−1, 1) with respect to the weight function w(x) = (1 − x2)−1/2. Starting with T0(x) = 1 …

The Chebyshev polynomials are both orthogonal polynomials and the trigonometric cos nx functions in disguise, therefore they satisfy a large number of useful relationships.

In the course of writing his doctoral dissertation, CHEBYSHEV considered the question of the distribution of prime numbers. The motivation for this came from the number-theoretic work of …

The Chebyshev equation is of particular interest in applications, because for particular choices of the parameter x its solutions generate an orthogonal sequence of polynomials, which satisfy min-max …

Summary Chebyshev is regarded as the founder of the St. Petersburg School of mathematics, which encompassed path-breaking work in probability the-ory. The Chebyshev Inequality carries his name; …

Theorem (Roots of Chebyshev polynomials) The roots of Tn(x) of degree n 1 has n simple zeros in [ xk = cos 2k 1 2n ; for each k = 1; 2 n : Moreover, Tn(x) assumes its absolute extrema at x0 = cos k k ; …