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Bayesian analysis is a statistical procedure which endeavors to estimate parameters of an underlying distribution based on the observed distribution. Begin with a "prior ...

Any real function u(x,y) with continuous second partial derivatives which satisfies Laplace's equation, del ^2u(x,y)=0, (1) is called a harmonic function. Harmonic functions ...

J_m(x)=(2x^(m-n))/(2^(m-n)Gamma(m-n))int_0^1J_n(xt)t^(n+1)(1-t^2)^(m-n-1)dt, where J_m(x) is a Bessel function of the first kind and Gamma(x) is the gamma function.

The Jacobi theta functions are the elliptic analogs of the exponential function, and may be used to express the Jacobi elliptic functions. The theta functions are ...

Let (q_1,...,q_n,p_1,...,p_n) be any functions of two variables (u,v). Then the expression ...

For a single variate X having a distribution P(x) with known population mean mu, the population variance var(X), commonly also written sigma^2, is defined as ...

A variable x is memoryless with respect to t if, for all s with t!=0, P(x>s+t|x>t)=P(x>s). (1) Equivalently, (P(x>s+t,x>t))/(P(x>t)) = P(x>s) (2) P(x>s+t) = P(x>s)P(x>t). (3) ...

The ratio of two independent estimates of the variance of a normal distribution.

The partial differential equation u_(xy)+(alphau_x-betau_y)/(x-y)=0.

J_m(x)=(x^m)/(2^(m-1)sqrt(pi)Gamma(m+1/2))int_0^1cos(xt)(1-t^2)^(m-1/2)dt, where J_m(x) is a Bessel function of the first kind and Gamma(z) is the gamma function. Hankel's ...