Opmerkingen en berekeningen betreffende "Poisson-proces met interrupties" van J. C. A. Zaat

Summary
In an earlier paper by J.C. A. ZAAT an interrupted Poisson process was studied. Here we clarify some of the results there obtained. The process is obtained from a stationary Poisson process on the real axis by covering the axis with a sequence of adjoining intervals which have alternatively length a and b, the first left-hand endpoint of an a-interval to the right of 0 being chosen in a stochastic point with If has a rectangular distribution on [0, a + b] the interrupted Poisson process is stationary, but the distribution of the length of an interval between two successive points in the interrupted process is now different for different intervals. For each distribution ofy over [0, a + b] the distribution of the length of the interval between the nth and the (n + I)st point to the right ofO in the interrupted Poisson process tends to the distribution function G(y) as n tends to infinity. Somehow ZAAT based his calculations exclusively on this stationary distribution.     

Asymptotic normality in vacancies on a line

Abstract  Covering a row of hooks by hats (one hat needs two adjacent hooks) by choosing for the next hat a pair of free hooks with equal probability from the free hooks, we encounter x_n , the number of isolated hooks remaining uncovered. We prove that x_n is asymptotically normal.     

Mean, median, mode

Summary This note is an attempt to avoid doing the same search for the third time. It happened twice in my life that I wished to prove that the median is located between mean and mode for certain B -distributions: first in 1953, next in 1976. For arbitrary distributions the result is sometimes referred to as F echner's theorem. Of course it does not hold in general. In order to prove the result for particular distributions one can often use an elegant theorem of T imerding . There is a nice relationship with the standardized third central moment.     

Discrete Spacings

Consider a string of n positions, i.e. a discrete string of length n . Units of length k are placed at random on this string in such a way that they do not overlap, and as often as possible, i.e. until all spacings between neighboring units have length less than k . When centered and scaled by n ^−1/2 the resulting numbers of spacings of length 1, 2,…,  k −1 have simultaneously a limiting normal distribution as n →∞. This is proved by the classical method of moments.     

Some remarks concerning the quotient of sample median and sample range for a sample of size 2n+1 from a normal distribution

Consider an ordered sample ₍₁₎, ₍₂₎,…, _(2n+1) of size 2 n +1 from the normal distribution with parameters μ and . We then have with probability one
₍₁₎ < ₍₂₎ < … < (2 n +1).
The random variable
_n =_(n+1)/_(2n+1)-(1)
that can be described as the quotient of the sample median and the sample range, provides us with an estimate for μ/, that is easy to calculate. To calculate the distribution of h _n is quite a different matter***. The distribution function of h₁, and the density of h₂ are given in section 1. Our results seem hardly promising for general h_n. In section 2 it is shown that h_n is asymptotically normal.
In the sequel we suppose μ= 0 and = 1, i.e. we consider only the "central" distribution. Note that h_n can be used as a test statistic replacing Student's t. In that case the central h_n is all that is needed.