Formules relatives aux probabilités qui dépendent de très grands nombers

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1 Formles relatives ax probabilités qi dépendent de très grands nombers M. Poisson Comptes rends II (836) pp In the most important applications of the theory of probabilities, the chances of events are expressed by some fractions which have for nmerator and for denominator some prodcts of a great nmber of neqal factors; this which renders the calclation of these fractions completely impractical, either directly or by aid of logarithms. One is then obliged to recr to certain formlas of approximation of which Stirling has given the first example, that Eler has next considered, and that Laplace has made depend on a method of redction in series proper to the qantities which he has named generally the fnctions of large nmbers. These formlas have this of the singlar that they contain the ratio of the circmference to the diameter, the base of the Naperian logarithms and other transcendents, which enter ths into the approximate vales of qantities of which the exact vales wold be whole nmbers or of ratios of similar nmbers. Their sage is especially indispensable in the qestions which have for object the chances of some ftre events dedced from observations of past events, that is to say in the most nmeros qestions of the calcls of probabilities; becase it is rare that we know a priori the chances of events, and if one excepts the most simple games where it is possible to enmerate the favorable cases and the cases contrary to each event, we are nearly always obliged to sbstitte in this enmeration the knowledge of the nmbers of times that the diverse events have taken place in some very great nmbers of trials. Bt in this regard one mst remark that the known rles of the theory of probabilities, and for example the theorem of Jacob Bernolli, spposes implicitly that the chance of each event is the same in the series of trials already made and in the ftre trials, while on the contrary this chance varies the most often in an nknown and completely irreglar manner, as well in the things of physical order as in those of the moral order. I have therefore soght to extend the rles of which there is concern to the general case of continally variable chances, and it is this which has led me to the demonstration of the law of large nmbers, which one will find in the work of which I actally occpy myself. This law consists in this that if all the possible cases, knowns or nknowns, either of the arrival of an event, or of the magnitde of a thing, remain constantly the same in many seqences of a very great nmber of trials, whatever be besides the nmber and the natre of these cases, the ratio of the nmber of times that the event will arrive, to the nmber of experiences, ths as the sm of the magnitdes of Translated by Richard J. Plskamp, Department of Mathematics & Compter Science, Xavier University, Cincinnati, OH. Janary 2, 200

2 the thing which will be observed, divided by the nmber of observations, will remain also very nearly the same in the different seqences. In order to leave nothing vage and ncertain in the expression of this fndamental law, it is necessary to determine the probability that the difference between the reslts of two seqences of operations will be contained between some given limits; of sch kind that if this probability neared mch to certitde, and if the experience gives nevertheless a difference which exits from these limits, one is jstified to conclde from it that the nknown cases of the events have changed in the interval of the two seqences. It is necessary ths to explicate, in a precise manner, this which one nderstands here by some cases which remain constantly the same. Now, when the arrival of an event or the magnitde of a thing are able to be attribted to different cases in any nmber, each of them has a determined probability, known or nknown, and gives to this arrival or to this magnitde a determined chance that one is able also to know or not know. This pt, we say that a case, whatever be its natre, is remained the same, when its particlar probability, and the chances of the event, if its probability was certain, have not experienced any change. One makes besides no hypothesis on the magnitdes of this chance and of this probability; one eliminates them both, and the definitive formlas contain only some nmbers given immediately by observations. It is for this, as I have said on nmeros occasions, that these formlas arrive indistinctly to the things of each natre, physical or moral. In order to rise against their application to all the cases a difficlty which merited some attention, it wold be necessary to show that their demonstration wold not be satisfied, or else it wold be necessary to cite some cases where the conseqences which are dedced from it wold have been denied by experience, that is to say some examples where the ratios which mst be very nearly constants, wold have varied notably, althogh no change had occrred in the cases of the events. Here is now a set of formlas ssceptible of freqent and varied applications. Many are new; others have already been given in my preceding Memoirs, or were known before; all are demonstrated in a chapter in my work, where the delicate analysis on which they depend is exposed with all the necessary developments. The nmber of trials, spposed very great, is represented by ; it is composed of two parts m and n which are spposed also very great nmbers; the formlas are so mch more close as this nmber is more considerable; and they wold be completely exact if were infinite. I. Let p and q be the constant chances dring all the dration of the trials, of two contrary events E and F, so that one has p + q =. We name U the probability that in the nmber or m + n trials, E will arrive m times and F will arrive n times. One will have U = ( p m ) m ( q ) n n 2πmn π designating as ordinarily the ratio of the circmference to the diameter. This formla is redced to U = e ν2 2π pq () 2

3 when one takes m = p ν 2 pq, n = q + ν 2 pq; ν being a given qantity, positive or negative, bt very small with respect to, and e designating the base of the Naperian logarithms. And nder this form, the expression of U sbsists eqally when the chances of E and F vary from one trial to another, by taking then for p and q the means of their vales in the entire series of sccessive trials. II. The events E and F having taken place effectively m and n times in the effective trials, and their constant chances p and q being nknowns, let U be the probability that they will arrive in or m + n ftre trials, of the nmber of times m and n proportionals to m and n, or sch that one has m = m n = n. Whatever be the nmber, one will have U = + U (2) by representing by U the probability of the ftre event which will take place if the ratios m and n were certainly the chances of E and F, that is to say, by making for brevity ( ) m m ( ) n n m n = U. III. The chances p and q of E and F were given, let P be the probability that in or m + n trials, E will arrive at least m times and F at most n times. One will have P = e t2 dt + ( + n) 2 π k 3 π pq e k2, P = e t2 dt + ( + n) (3) 2 π 3 π pq e k2 ; k being a positive qantity of which the sqare is k 2 n = nlog q( + ) k + (m + )log m + p( + ), where the logarithms are Naperian; and by employing the first or the second formla, according as one will have q p > m+ n, or q p < m+ n. IV. By naming R the probability that E and F will take place in trials, of the nmber of times which will not exit from the limits p 2 pq, q ± 2 pq, Translator s note: At this point, Poisson introdces the prime as both sbscript and sperscript. The typesetter seems to have had difficlty with them. I have preserved their locations as it appears in the text rather than to attempt corrections. This problem occrs throghot. 3

4 where is a positive and given qantity, bt very small with respect to, one will have R = 2 e t2 dt + e n2 ; (4) π 2π pq and reciprocally if the chances p and q are nknowns, and if E and F are arrived the nmber of times m and n in or m + n trials, one will have R = 2 e t2 dt + π 2πmn e 2 ; (5) for the probability that the vales of p and q will not exit from the limits m ± 2mn, n 2mn. n V. In two different seqences of large nmbers and of trials, let m and m be the nmbers of times that E has taken place or will take place, n and n the nmber of times that F will arrive or is arrived; we designate by a positive qantity, very small with respect to and ; and let ϖ be the probability that the difference m m will not exit from the limits 2( 3 m n + 3 m n ), no more than the difference n n, from these same limits taken with the contrary signs. One will have ϖ = 2 e t2 dt + π 2πm n ( + ) e 2 ( 3 m n + 3 mn) 3 m n (+ ) (6) As one will have also very nearly m = m and n = n, one will be able withot altering sensibly the vale of ϖ, to replace in its last term, what will be always a small fraction, the letters, m, n,by,m, n, and reciprocally those here by those there. This formla, by setting aside at least its last term, will agree in the general case where the chances of E and F will vary from one trial to another, provided that, in the two seqences, of possible cases, of events, known or nknown, experience no change, that is to say, provided that each of these cases, ths as one has said above, conserve the same probability and give always the same chance to the arrival of E or of F. VI. The nmber of times that E and F are arrived in the trials relative to these events, being always m and n, are generally m and n, the nmber of times that two other contrary events E and F have taken place in a nmber trials, also very great. We sppose that one has m m = δ; δ being a small positive or negative fraction. We call p and p the nknown and spposed constant chances of E and E ; and we designate by Q the probability that p 4

5 exceeds p by a small positive and given fraction ε. By representing by a positive qantity, and making = ± (ε δ) 2( 3 m n + 3 mn), according as the factor ε δ will be positive or negative, one will have Q = e t2 dt, Q = e t2 dt (7) π π the first expression is retrning to the case where the difference ε δ will be positive, and the second in the case where this difference will be negative. The same formlas will express also the probability that the nknown chance p of the arrival of E srpasses given by observation, of a fraction ω also given. For this, it will sffice to the ratio m make = ± ( ω m ) 2mn, and to take the first or the second formla, according as the difference ω m will be positive or negative. VII. When the chances of the two contrary events E and F vary from one trial to another, let p i and q i be their vales relative to the trial, of which the rank is marked by i, so that one has p i + q i =, for all the indices i. The sms extending from i = to i =, we make, for brevity, p i = p, q i = q, 2 p i q i = k 2. Let always m and n be the nmber of times that E and F are arrived in the first trials. By designating by a positive and given qantity, very small with respect to, one will have R = 2 e t2 dt + π k π e 2, (8) for the probability that the ratios m and n will not exit from the limits p k, q ± k this which coincides with formla (4) in the particlar case of constant chances. VIII. Any thing A being ssceptible to all the vales contained between the limits h g, all these vales being eqally possible and the only possible, let P be the probability that in any nmber i of trials, the sm of the vales of A which wold take place, will be contained between the limits given also c ε. One will have 2(2g) i P = Γ Γi i, (9) 5

6 by making, for brevity, r = ± (ih + ig c + ε) i i(ih + ig 2g c + ε) i ± i.i.2 (ih + ig 4g c + i.i.i 2 ε)i (ih + ig 6g c + ε) i ± etc.,.2.3 r = ± (ih + ig c ε) i i(ih + ig 2g c ε) i ± i.i.2 (ih + ig 4g c ε)i i.i.i 2 (ih + ig 6g c ε) i ± etc.,.2.3 by taking, in each term, the sperior sign or the inferior sign, according as the qantity which is fond raised to the power i is positive or negative: g and ε are some positive qantities, h and c are able to be some positive or negative qantities. IX. Whatever be the law of probability of the possible vales of the thing A in each trial, and the manner of which this law will vary from one trial to another, if one calls s the sm of the vales of A which will take place in a very great nmber of trials, one will have P = 2 e t2 dt, (0) π for the probability that the mean s of the vales of A will fall between the limits k 2 h ; designating a positive qantity and very small with respect to ; k and h being some qantities of which the second is positive, and which depends on the probabilities of the vales of A dring all the dration of the trials. When these probabilities will be constants, eqals for all the possible vales between the given limits a and b, and nll beyond these limits, one will have k = (a + b), 2 h = b a 2 6. When A will have only a finite nmber of possible vales c, c 2, c 3,...c ν, and when these constant vales will be all eqally probable, one will have k = ν (c + c 2 + c 3 + c ν ), h = 2ν 2 [ ν ( c 2 + c c c 2 ν) (c + c 2 + c 3 + c ν ) 2]. X. Let λ n be the vale of A which takes place at the n th trial; and we make λ n = λ, (λ n λ) 2 = 2 l2 ; the sms extending from n = to n =. We sppose that the cases of all the possible vales of A experience no change, either in their respective probabilities, or 6

7 in the chances which they give to each of these vales. There will be then a special qantity γ of which the mean s of the vales of A will approach indefinitely in measre as will increase more and more, and that it will attain, if became infinite. Now, formla (0) will express the probability that this qantity γ is contained between the limits s l, which contain nothing nknown. XI. In a second series of a very great nmber of trials, let s be the sm of the vales of A, and l that which the qantity l will become which is related in the first series. Formla (0) will express eqally the probability that the difference s s of the two means will be contained between the limits s ± l l 2 + l 2, or else, becase one will have very nearly l = l, this will be the probability that the mean s, relative to the second series, will fall between the limits s ± l +, which depend only on the reslts of the first and on the given qantity, and which are so mch more narrow as is greater with respect to. XII. In order to determine the vale of one same thing A, one has made many seqences of trials, which comprehend very great nmbers,,, etc. The sms of the vales of A, which one has obtained in these sccessive seqences, are s, s, s,etc.; the preceding qantity l is related always to the first series, and one designates by l, l, etc., this which it becomes in regard of the following seqences. One spposes that the cases of errors in the measres vary from one seqence to another, bt that nevertheless all the means s, s, s, etc., converge indefinitely in measre as,,, etc., increase more and more, toward one same nknown qantity γ, which wold be the tre vale of A, in the most ordinary case where these cases do not render neqally probable, in any of these seqences of observations, the errors eqal and of contrary signs. This pt, formla (0) will express frther the probability that the qantity γ is contained between the limits: in which one makes, for brevity, sq + s q + s q + etc. D, l l 2 l 2 + etc. = D2, D 2 l 2 = q, D 2 l 2 = q, D 2 l 2 = q, etc. 7

8 Moreover, the part sq + s q + s q +etc., that is to say the sm of the means s, s, s, etc., mltiplied respectively by the qantities q, q, q, etc., will be the approximate vale of γ, the most advantageos that one is able to dedce from the concrrence of all the seqences of observations, that is to say the vale of this nknown of which the limits of error D will have the least extent that is possible for a given vale of, or else, in eqal degree of probability. XIII. Finally, the cases of the arrival of an event E remain the same dring the trials, the ratio m of the nmber of times that E will take place to the total nmber of trials, will converge indefinitely toward a special qantity r, which it wold attain rigorosly, if became infinite. Now, formla (6), by neglecting its last term, or else again formla (0), will be the probability that the nknown vale r falls between the limits m ± 2m( m). One finds at the end of the Analyse des réfractione astronomiqe of Kramp, a table of nmerical vales of n e t2 dt, which extends from = 0 to = 3. This integral decreases very rapidly when the vale of increases; it is eqal to 2 π, for = 0, and its vale falls below two hndred thosandths, for = 3. If one takes = , one will have, very nearly, 2 e t2 dt = π 2 ; this which will redce also to 2 the probability expressed by formla (0). By giving to a vale sch that = 5 or = 4, which withot being considerable renders extremely small that of the integral e t2 dt, the reslts which one jst annonced contains the law of large nmbers in all its generality. 8