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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. 相似文献
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. 相似文献
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J.Th. Runnenburg 《Statistica Neerlandica》1985,39(2):181-189
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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. 相似文献
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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 xn , the number of isolated hooks remaining uncovered. We prove that xn is asymptotically normal. 相似文献
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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 hn 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 hn . In section 2 it is shown that hn is asymptotically normal.
In the sequel we suppose μ= 0 and = 1, i.e. we consider only the \"central\" distribution. Note that hn can be used as a test statistic replacing Student's t. In that case the central hn is all that is needed. 相似文献
The random variable
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
In the sequel we suppose μ= 0 and = 1, i.e. we consider only the \"central\" distribution. Note that h
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We know that the partial means mrof a sequence of i.i.d. standardized random variables tend to 0 with probability 1. If we want P{mk≥εfor some k ≥r}≤δ for given positive ε and δ, how large should we take r? Several (strong) inequalities for the distribution of partial sums providing an answer to this question can be found in the literature (Hájek -Rényi Robbins , Khan ). Furthermore there exist wellknown (weak) inequalities (Bienaymé -Chebyshev , Bernstein , Okamoto ) that give us values of rfor which P{mr≥ε}≤δ. We compare these inequalities and illustrate them with numerical results for a fixed choice ofε and δ. After a general survey and introduction in section 1, the normal and the binomial distribution are considered in more detail in the sections 2 and 3, while in section 4 it is shown that the strong inequality essentially due to Robbins can give an inferior result for particular distributions. 相似文献