Gunnar Blom

HOW MANY RANDOM DIGITS ARE REQUIRED UNTIL GIVEN SEQUENCES ARE OBTAINED

Random digits are collected one at a time until a given k-digit sequence is obtained, or, more generally, until one of several k-digit sequences is obtained. In the former case, a recursive formula is given, which determines the distribution of the waiting time until the sequence is obtained and leads to an expression for the probability generating function. In the latter case, the mean waiting time is given until one of the given sequences is obtained, or, more generally, until a fixed number of sequences have been obtained, either different sequences or not necessarily different ones. Several results are known before, but the methods of proof seem to be new.

Extrapolation of Linear Estimates to Larger Sample Sizes

Abstract A study is performed of McCools method for constructing linear estimates from order statistics when the coefficients of a linear estimate for a smaller sample size are available. The efficiency of the method is investigated, using the theory of U statistics. Among other things, it is shown that in the case of the logistic distribution, the method produces an asymptotically efficient location parameter estimate and, in the case of the Pareto distribution, an asymptotically efficient scale parameter estimate.

Random Walks of Ordered Elements with Applications

Abstract In this partly expository paper a simple random walk model is introduced for generating sequences of ordered elements. The model is related to Polyas urn scheme. An alternative description of the walk leads to conditionally independent steps. A basic theorem for the walk is proved and applications are given. Among other things, recursion formulas for moments of order statistics are studied and connections with exceedances and distribution-free tolerance intervals are indicated.

The Mississippi Problem

Abstract We present a tricky combinatorial problem, primarily intended for entertainment. Two more problems are given as a challenge to the reader at the end of the article.

Some properties of similar pairs

In a given set, the elements are compared pairwise. The number W of similar pairs is studied, that is, the number of pairs with a certain property in common. Under certain conditions, W has, approximately, a Poisson distribution. Examples are considered connected with the birthday problem and with a circle problem involving DNA breakages.

On the stochastic ordering of waiting times for patterns in sequences of random digits

Random digits are collected one at a time until a pattern with given digits is obtained. Blom (1982) and others have determined the mean waiting time for such a pattern. It is proved that when a given pattern has larger mean waiting time than another pattern, then the waiting time for the former is stochastically larger than that for the latter. An application is given to a coin-tossing game. COIN-TOSSING GAME; GENERATING FUNCTION; LEADING NUMBERS; LEADING QUANTITY; OVERLAP

When is the Arithmetic Mean Blue

In a research report [ 1], Andreasson, apparently inspired by Hammersley and Handscombs book on Monte Carlo methods [2], discussed how to best combine results of repeated Monte Carlo calculations when antithetic variables are used. Some of his results have general statistical interest and are presented here with some extensions, examples and comments. In ?2, some known results concerning best linear unbiased estimates are briefly reviewed. In ?3, the special simulation problem in [1] is presented. In ?4, the solution is given in the form of two general theorems. Some other applications are given.

The distribution of the record position and its applications

Abstract This is mainly an expository article on the positions of records in sequences of ordered elements. Such sequences are obtained, for example, when observing and ordering continuous iid random variables. In practice, records are of interest, for example, in meteorology and sports. A k-record is obtained when a new element is placed at position k counted from the top. The sequences of time points, when new k-records occur, are studied by elementary random walk methods. In the last section, it is shown that the time scale can be changed so that the time points of the k-records follow, approximately, a Poisson process.

Basic probability theory I

In this chapter, we visit the vaguely defined area of basic probability theory; in our conception of the world of probability this area includes elementary theory for rv’s. Ideas are put forward concerning conditional probability, exchangeability, combination of events and zero—one rv’s. Do not forget the last section about the ‘zero—one idea’, which provides a powerful tool in many otherwise awkward situations. A good general reference for basic discrete probability is Feller (1968); another is Moran (1968).

The Binomial and Related Distributions

The binomial distribution was introduced in §3.5, where we also mentioned the hypergeometric distribution, the Poisson distribution and the multinomial distribution. All these distributions are related to the binomial distribution. In the present chapter, we will discuss the binomial distribution in §9.2, the hypergeometric distribution in §9.3, the Poisson distribution in §9.4, and the multinomial distribution in §9.5.