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

structure of statistical methods, discuss what is certain and whattentative there and on what premises are they founded.1.2. The part played by mathematical models. Any statisticaltreatment must be preceded by a mathematical model of thephenomenon studied stating which magnitudes are random, which not;which are dependent, and which not, etc. Sometimes you willencounter a delusion that tells you that if any magnitude is notdeterminate (if its values can not be precisely predicted), it may beconsidered random. This is completely wrong because randomnessdemands statistical stability. Therefore, indeterminate behaviour is notgenerally speaking, randomness; or, if you wish, in addition todeterminate and random there exist indeterminate magnitudes whichwe do not know how to deal with.A mathematical model can include either determinate or randommagnitudes, or both, but, as of today, not those last mentioned. The artof choosing a mathematical model therefore consists in approximatelyrepresenting the indeterminate magnitudes appearing practicallyalways as either determinate or random. It is also necessary that thevalues of the determinate magnitudes or the distributions of theprobabilities of the random variables be derivable from theexperimental material at hand (or available in principle).Let us return to the determination of the efficacy of a newpreventive measure. We have an observational seriesµ 1 , µ 2 , ..., µ n , µ (1.1)where µ i are the numbers of failures for the previous years and µ , thesame for the year when the innovation is being tested. Where is themathematical model here? In case of rare failures it is ratherreasonable to assume that the series (1.1) is composed of randomvariables. However, when introducing that term, we oblige ourselvesto state the statistical ensemble of experiments in which the variable isrealized. Two paths are open: either we believe that the number offailures before the innovation was implemented are realizations of arandom variable, or we imagine the results of many sets of machineryidentical to our set working under the same conditions. In the first, butnot necessarily in the second case the magnitudesµ 1 , µ 2 , ..., µ n (1.2)ought to be identically distributed. Or, assuming a Poisson distribution,we have in the first caseEµ 1 = Eµ 2 = ... = Eµ n = λ (1.3)and, in the second case we may assume thatEµ 1 = λ 1 , Eµ 2 = λ 2 , ..., Eµ n = λ nwhere49