unitBall_normGauss-class {multIntTestFunc}R Documentation

An S4 class to represent the function 1(2π)n/2exp(x22/2)\frac{1}{(2\pi)^{n/2}}\exp(-\Vert\vec{x}\Vert_2^2/2) on BnB^{n}

Description

Implementation of the function

f ⁣:Bn[0,),xf(x)=1(2π)n/2exp(x22/2)=1(2π)n/2exp(12i=1nxi2),f \colon B_n \to [0,\infty),\, \vec{x} \mapsto f(\vec{x}) = \frac{1}{(2\pi)^{n/2}}\exp(-\Vert\vec{x}\Vert_2^2/2) = \frac{1}{(2\pi)^{n/2}}\exp(-\frac{1}{2}\sum_{i=1}^n x_i^2),

where n{1,2,3,}n \in \{1,2,3,\ldots\} is the dimension of the integration domain Bn={xRn:x21}B_n = \{\vec{x}\in R^n : \Vert \vec{x} \Vert_2 \leq 1\}. In this case the integral is know to be

Bnf(x)dx=P[Z1]=Fχn2(1),\int_{B_n} f(\vec{x}) d\vec{x} = P[Z \leq 1] = F_{\chi^2_n}(1),

where ZZ follows a chisquare distribution with nn degrees of freedom.

Details

The instance needs to be created with one parameter representing nn.

Slots

dim

An integer that captures the dimension

Author(s)

Klaus Herrmann

Examples

n <- as.integer(3)
f <- new("unitBall_normGauss",dim=n)

[Package multIntTestFunc version 0.2.0 Index]