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If x takes only nonnegative values, then P(x>=a)<=(<x>)/a. (1) To prove the theorem, write <x> = int_0^inftyxP(x)dx (2) = int_0^axP(x)dx+int_a^inftyxP(x)dx. (3) Since P(x) is ...

The equation f(x_n|x_s)=int_(-infty)^inftyf(x_n|x_r)f(x_r|x_s)dx_r which gives the transitional densities of a Markov sequence. Here, n>r>s are any integers (Papoulis 1984, ...

A type I Markov move.

A type II Markov move.

A theorem proved by Doob (1942) which states that any random process which is both normal and Markov has the following forms for its correlation function C_y(tau), spectral ...

A stochastic matrix, also called a probability matrix, probability transition matrix, transition matrix, substitution matrix, or Markov matrix, is matrix used to characterize ...

Apply Markov's inequality with a=k^2 to obtain P[(x-mu)^2>=k^2]<=(<(x-mu)^2>)/(k^2)=(sigma^2)/(k^2). (1) Therefore, if a random variable x has a finite mean mu and finite ...

A spectrum formed by the Lagrange numbers. The only ones less than three are the Lagrange numbers, but the last gaps end at Freiman's constant. Real numbers larger than ...

The end of the last gap in the Lagrange spectrum, given by F=(2221564096+283748sqrt(462))/(491993569)=4.5278295661... (OEIS A118472). Real numbers greater than F are members ...

A spatial point process is a point process which models data that is localized at a discrete set of locations in space or, more specifically, on a plane.