A Note Concerning the Distances of Uniformly Distributed Points From the Centre of a Rectangle

Abstract

Given a rectangle containing uniformly distributed random points, how far are the points from the rectangle's centre? In this paper we provide closed-form expressions for the cumulative distribution function and probability density function that characterise the distance. An expression for the average distance to the centre of the rectangle is also provided.2010 Mathematics subject classification: primary 60D05; secondary 68T40.

A library to compute the density of the distance between a point and a random variable uniformly distributed in some sets

“…We start with an example written in file drectex.m of folder Examples where S is a rectangle with side lengths L and αL with 0 < α < 1 and P is the center of the rectangle. For this example, the density of the distance from P to a random variable uniformly distributed in S is known in closed form and is given in [5]. Therefore, this example allows us to test the implementation of function density polyhedron comparing output d of this function when algo='g', 't1' with the theoretical values given in [5].…”

“…For this example, the density of the distance from P to a random variable uniformly distributed in S is known in closed form and is given in [5]. Therefore, this example allows us to test the implementation of function density polyhedron comparing output d of this function when algo='g', 't1' with the theoretical values given in [5].…”

“…, x Np are Np equally spaced points in ]dmin,dmax[. • d is a vector of size Np: d(i) is the exact value of the density at x i computed using the analytic formulas given in [5]. • ErrG is the maximal error when algo='g', i.e., ErrG= max i=1,...,Np |dG(i) − d(i)|.…”

Optical Wireless Communication Based Indoor Positioning Algorithms: Performance Optimisation and Mathematical Modelling

“…There exists a large amount of research to find the standard deviation of uniformly distributed random points around the centre of a circle, rectangle or a square [53][54][55][56], which gives closed form expressions for the second raw moment of these geometrical shapes. In this paper, these closed form expressions are used to find the standard deviation for different values of beam radius using the normalised second raw moment with respect to area of the shape, and then compared with simulation results.…”

“…For a rectangular beam shape with length h and breadth b, standard deviation σ r can be expressed as [54][55][56][57]:…”