1 Studies in the History of Statistics and Probability ... - Sheynin, Oscar

which means that sooner or later the scatter of the values of E m (X)rather often appreciably diminishes. The author introduced the pattern of anextended series of trials. Bearing in mind his statements made in the sequel, it meansthat the behaviour of E m (X) is studied throughout the series rather than appreciatedby the result of the last trial. This latter method is called the pattern of a fixed series.For a predictable permanence it is supposed that (1) persists when the series isextended and E(X) is the predictable rough estimate of the empirical mean.Expectation of a separate measurement is meaningless.The author introduces moments but barely applies them.2.2. Statistical stability. It is often alleged that homogeneity oftrials leads to statistical stability. Only controlled conditions of trialsare meant and therefore, on the contrary, statistical stability means thatthe trials were homogeneous. Statistical stability is best justified byempirical induction. Without stability E(X) does not exist.Randomness (in the general sense) is identified withunpredictability. It became usual to understand random variables in themathematical sense only as statistically stable unpredictablemagnitudes, and even such for which the notion of distribution ofprobabilities is applicable.This narrow specialized interpretation of random variable is stillbeing willy-nilly confused with its wide general meaning and leads toa mistaken belief that the applied theory of probability andmathematical statistics are applicable to any random variableunderstood in the general sense, i. e., to unpredictable magnitudes.On the other hand, the reader begins to believe that themathematical propositions of the theory of probability somehowdirectly concern only such magnitudes. Actually, their unpredictabilityis not at all a necessary condition for applying to them the theory ofprobability. It is important that when measuring a magnitude manytimes it indicates statistical stability. An artificial introduction ofunpredictability in an experiment by the so-called randomization asalso in some calculations by the Monte Carlo method can be thought tomean an excessively brave challenge to the natural scientific tradition 1 .2.3. Probability of an event. An event A is random in both senses if X A (s)is random. Stability of frequency is established by empirical induction according tothe pattern of an extended series. If frequency is stable, E(X), the probability of anevent, is its predictable rough estimate. If the behaviour of the series is not studied,and the probability only determined by its outcome, the statistical stability is notinvestigated.Statistical probability is not applicable to individual trials. For estimating theprobability of a rare event of the order of 10 −4 , sometimes encountered in thereliability theory, 10 5 measurements are required.2.4. Distribution of probabilities. It is measured for a series of anincreasing number of trials. If the empirical distributions are stabilized, F(X) isdetermined. This is empirical induction for the pattern of an extended series. Lack ofstability of the empirical distributions means that the notion of F(X) is not applicable.Often recommended is the measurement of those F m (X) because their stability ismore noticeable, rather than the histograms, but this is akin to stating that aninsensitive device is better than a sensitive histogram (indicating a greater scatter, alack of stability).2.5. Statistical independence. Lack of correlation. A necessaryand sufficient condition of independence isF(X 1 , X 2 , ..., X n ) = F 1 (X 1 ) F 2 (X 2 ) ... F n (X n ).125