But <strong>the</strong>n <strong>the</strong> newly appear<strong>in</strong>g peculiarities <strong>of</strong> that material will havenot been noticed.What k<strong>in</strong>d <strong>of</strong> peculiarities could happen? For example, on Fig. 2a<strong>and</strong> 2b a certa<strong>in</strong> non-zero number <strong>of</strong> observations is shown <strong>in</strong> <strong>the</strong>region µ* ≤ − 3. The probability <strong>of</strong> one observation be<strong>in</strong>g <strong>the</strong>re(assum<strong>in</strong>g that <strong>the</strong> Mendelian laws are valid) is 0.0014, <strong>and</strong>, <strong>of</strong> one out<strong>of</strong> approximately a hundred (to recall, <strong>the</strong> numbers <strong>of</strong> observationswere 98 <strong>and</strong> 123), about a hundred times higher; here, almost preciselyso. Thus, <strong>the</strong> probability <strong>of</strong> observations appear<strong>in</strong>g <strong>in</strong> that region <strong>in</strong>both series is about 0.14 2 ≈ 0.02, which means that a deviation fromnormality N(0, 1) is significant on <strong>the</strong> level ≈ 2%. So, are <strong>the</strong>Mendelian laws never<strong>the</strong>less wrong? Well, first <strong>of</strong> all, we have chosena test correspond<strong>in</strong>g to known data; second, a perfectly reasonableattitude does not mean dogmatically follow<strong>in</strong>g tests <strong>of</strong> significance. Areasonable answer apparently means that <strong>the</strong> bulk <strong>of</strong> observationsperfectly agrees with <strong>the</strong> Mendelian laws but sharp deviations perhapsdo occur. It can be supposed that a deficiency <strong>in</strong> <strong>the</strong> number <strong>of</strong>displayed recessive <strong>in</strong>dications has some biological sense (if,accord<strong>in</strong>g to a very simple explanation, <strong>the</strong>re exists a connection withsurvivorship).Incidentally, <strong>the</strong> above sufficiently illustrates <strong>the</strong> simple idea thattruth <strong>in</strong> science is established by <strong>the</strong> work <strong>of</strong> a number <strong>of</strong> generations<strong>and</strong> is not always atta<strong>in</strong>ed at each separate <strong>in</strong>vestigation.2.2. No one knows <strong>the</strong> hour ... The ancient say<strong>in</strong>g, No one knows<strong>the</strong> hour <strong>of</strong> his death, became somewhat shaken (certa<strong>in</strong>ly <strong>in</strong> <strong>the</strong>statistical ra<strong>the</strong>r than <strong>in</strong>dividual sense) after life tables have beencompiled <strong>and</strong> it occurred that <strong>the</strong> probability <strong>of</strong> liv<strong>in</strong>g up to a def<strong>in</strong>iteage, is subject to fluctuations (depend<strong>in</strong>g on <strong>the</strong> conditions <strong>of</strong> life),which are, however, not too essential. A fur<strong>the</strong>r step towards an<strong>in</strong>dividual forecast based on multivariate statistical analysis is partlymade <strong>and</strong> partly be<strong>in</strong>g made. I am describ<strong>in</strong>g one <strong>of</strong> <strong>the</strong> mostoutst<strong>and</strong><strong>in</strong>g contributions <strong>in</strong> this field, <strong>the</strong> so-called Fram<strong>in</strong>gham<strong>in</strong>vestigation (one <strong>of</strong> <strong>the</strong> pert<strong>in</strong>ent publications is Truett et al 1967).The cardiovascular diseases are known to be one <strong>of</strong> <strong>the</strong> centralproblems <strong>of</strong> modern medic<strong>in</strong>e. They are manifested <strong>in</strong> different ways;one <strong>of</strong> <strong>the</strong> most common k<strong>in</strong>d is <strong>the</strong> so-called ischemic heart disease(IHD). Accord<strong>in</strong>g to <strong>the</strong> classification adopted <strong>in</strong> <strong>the</strong> cited work, itcomprises cases <strong>of</strong> myocardial <strong>in</strong>farction, coronary <strong>in</strong>sufficiency,ang<strong>in</strong>a pectoris <strong>and</strong> deaths occasioned by disturbances <strong>of</strong> <strong>the</strong> coronaryblood circulation. We know well enough that <strong>the</strong> IHD <strong>of</strong>ten affectspeople yet be<strong>in</strong>g <strong>in</strong> <strong>the</strong> prime <strong>of</strong> creative power which makes <strong>the</strong>problem especially acute.There exist some ra<strong>the</strong>r vague ideas on <strong>the</strong> part played by <strong>the</strong> factors<strong>of</strong> modern <strong>in</strong>dustrialized life <strong>in</strong> <strong>the</strong> development <strong>of</strong> <strong>the</strong> IHD (littlephysical activity, nervous-emotional stress, irrational diet, etc) <strong>and</strong>also by <strong>the</strong> possible <strong>in</strong>fluence <strong>of</strong> genetic factors. These ideas arecerta<strong>in</strong>ly extremely important but we would like to have, <strong>in</strong> addition togeneral (but <strong>in</strong>sufficiently clear <strong>and</strong> <strong>in</strong>completely proven) ideas someamount <strong>of</strong> scientific (i. e. trustworthy) <strong>in</strong>formation.That, perhaps not cover<strong>in</strong>g <strong>the</strong> entire problem, would provide areliable foundation for some practical conclusions. Important is, for106
example, <strong>the</strong> problem <strong>of</strong> <strong>the</strong> objectively established risk factors. To<strong>the</strong>se belong, on <strong>the</strong> one h<strong>and</strong>, portents <strong>of</strong> an illness established bymodern diagnostic means (e. g. changes <strong>in</strong> electro-cardiogram), on <strong>the</strong>o<strong>the</strong>r h<strong>and</strong>, factors <strong>of</strong> life <strong>and</strong> behaviour (age, smok<strong>in</strong>g, amount <strong>of</strong>cholesterol <strong>in</strong> <strong>the</strong> blood, etc). S<strong>in</strong>ce <strong>the</strong> bus<strong>in</strong>ess concerns someprecisely determ<strong>in</strong>ed factors ra<strong>the</strong>r than vaguely underst<strong>and</strong>ableexcessive tempo <strong>of</strong> modern life, a scientific <strong>in</strong>vestigation <strong>of</strong> <strong>the</strong>ir part is<strong>in</strong> pr<strong>in</strong>ciple not unlikely.The possible ways <strong>of</strong> <strong>the</strong> development <strong>of</strong> <strong>the</strong> IHD are little known,so <strong>the</strong> statistical method <strong>of</strong> study<strong>in</strong>g it is <strong>the</strong> ma<strong>in</strong> method. As usual,expectations here will be chiefly based on rely<strong>in</strong>g that a large amount<strong>of</strong> data will be able to compensate <strong>the</strong> deficiency <strong>of</strong> <strong>in</strong>formation about<strong>the</strong> essence <strong>of</strong> <strong>the</strong> phenomenon (<strong>in</strong> this case, <strong>of</strong> <strong>the</strong> IHD). And s<strong>in</strong>cethat disease develops gradually, over many years, it is desirable that<strong>the</strong> <strong>in</strong>vestigation covers not only a large number <strong>of</strong> people, but a verylong period <strong>of</strong> time as well (if possible, <strong>the</strong>ir whole life).A s<strong>in</strong>gle exam<strong>in</strong>ation <strong>of</strong> a large number <strong>of</strong> people presents seriousdifficulties; <strong>and</strong>, tak<strong>in</strong>g <strong>in</strong>to account that people usually move severaltimes dur<strong>in</strong>g <strong>the</strong>ir lifetime, you will underst<strong>and</strong> that <strong>the</strong> real difficultiesare great. It ought to be also borne <strong>in</strong> m<strong>in</strong>d that <strong>the</strong> relative number <strong>of</strong>cases (<strong>of</strong> people f<strong>in</strong>ally develop<strong>in</strong>g <strong>the</strong> IHD) is ra<strong>the</strong>r small, so that <strong>the</strong>population to be exam<strong>in</strong>ed mostly consists <strong>of</strong> non-cases (o<strong>the</strong>r people).Therefore, <strong>the</strong> loss <strong>of</strong> a non-case by <strong>the</strong> researcher is comparativelyunimportant, but los<strong>in</strong>g at least one case is extremely undesirable.However, if we allow <strong>the</strong> loss <strong>of</strong> people (for example, occasioned by<strong>the</strong> man’s move or refusal to come for <strong>the</strong> exam<strong>in</strong>ation), we do notknow whe<strong>the</strong>r it was a case or not <strong>and</strong> it should be attempted that <strong>the</strong>losses be as small as possible, so perhaps <strong>the</strong> greatest part <strong>of</strong> <strong>the</strong> entireeffort is spent to atta<strong>in</strong> that goal.The exam<strong>in</strong>ation covered practically <strong>the</strong> whole population <strong>of</strong> asmall American town Fram<strong>in</strong>gham aged 30 – 62 years at its beg<strong>in</strong>n<strong>in</strong>g.It is go<strong>in</strong>g on for more than 20 years <strong>and</strong> <strong>the</strong> cited source reports <strong>the</strong>results <strong>of</strong> <strong>the</strong> first twelve years. They are based on <strong>in</strong>vestigat<strong>in</strong>g 2187men <strong>and</strong> 2669 women not suffer<strong>in</strong>g <strong>in</strong>itially from <strong>the</strong> IHD. Itsdevelopment dur<strong>in</strong>g those twelve years was revealed <strong>in</strong> 258 men(11.8%) <strong>and</strong> 129 women (4.8%); it was known long ago that womensuffer from IHD more rarely than men.The connection between <strong>the</strong> risk factors measured dur<strong>in</strong>g <strong>the</strong> firstexam<strong>in</strong>ation <strong>and</strong> <strong>the</strong> probability <strong>of</strong> <strong>the</strong> development <strong>of</strong> <strong>the</strong> IHD dur<strong>in</strong>g<strong>the</strong> 12 follow<strong>in</strong>g years was considered. In general, it is possible to listra<strong>the</strong>r many such factors, but only seven were taken account <strong>of</strong>:1. Age (<strong>in</strong> years). 2. Content <strong>of</strong> cholesterol <strong>in</strong> <strong>the</strong> blood serum(mm/100 millilitre). 3. Systolic blood pressure (mm <strong>of</strong> mercurycolumn). 4. Relative weight (weight expressed <strong>in</strong> per cents <strong>of</strong> man’sweight relative to mean weight for appropriate sex <strong>and</strong> stature). 5.Content <strong>of</strong> haemoglob<strong>in</strong> (g/100 millilitre). 6. Smok<strong>in</strong>g (0, nonsmokers;1, 2 <strong>and</strong> 3, smok<strong>in</strong>g less than a packet daily, smok<strong>in</strong>g apacket <strong>and</strong> more than a packet). 7. Electro-cardiogram (0, normal, 1,abnormal).107
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Studies in the History of Statistic
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Introduction by CompilerI am presen
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(Lect. Notes Math., No. 1021, 1983,
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sufficiently securely that a carefu
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is energy?) from chapter 4 of Feynm
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demand to apply transfinite numbers
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for stating that Ω consists of ele
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chances to draw a more suitable apa
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Let the space of elementary events
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2.3. Independence. When desiring to
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Eξ = ∑ aipi.Our form of definiti
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absolutely precisely if the pertine
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where x is any real number. If dens
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probability can be coupled with an
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Nowadays we are sure that no indepe
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λ = λ(T)with λ(T) being actually
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(1/B n )(m − A n )instead of the
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along with ξ. For example, if ξ i
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µ( − p0) ÷np0 (1 − p0)nhas an
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distribution of the maximal term |s
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ξ (ω) + ... + ξ (ω)n1n{ω :|
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P{max ξ(t) ≥ x} = 0.01, 0 ≤ t
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1. This example and considerations
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IIV. N. TutubalinTreatment of Obser
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structure of statistical methods, d
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Suppose that we have adopted the pa
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and the variances are inversely pro
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