otes to accompany Continuatio argumenti de mensura sortis ad fortuitam successionem rerum naturaliter contingentium applicata Richard J. Pulskamp Department of Mathematics and Computer Science Xavier University, Cincinnati, Ohio March 006 This paper is, in some ways, the more interesting of the pair. Its topic is the comparison of two hypotheses concerning the ratio etween male and female irths whether data supports a : ratio or some other. However, in order to make this comparison, Bernoulli must develop methods of computing inomial proailities corresponding to very large sample sizes. However, Bernoulli, as he tests his theories upon tales of real data, is led to questions concerning correction of values and the staility of statistical ratios. Throughout his paper the numer of irths is denoted y and the ratio of male to a a+. female irths as a :. The proaility that a irth is male is then estimated as p = Let the random variale X have a inomial distriution with parameters and p and we will identify two particular functions that will e found useful in descriing the method of Bernoulli. Define the function q as q(m) = Pr(X = m p = /) = ( m ) ( ) and note that the maximum value of q(m) occurs where m =. This function q, which corresponds to the first hypothesis, will e seen to take on a kind of fiduciary role in his analysis. In the first paper, Bernoulli otained an estimate of q(). We note that q( ± µ) q()e µ /, a result which will ultimately e verified y Bernoulli. Similarly, we define r as r(m) = Pr(X = m p) = ( ) ( a ) m ( ) m m a +
a a+ /. This function corresponds to his second or alternative hypoth- where p = esis. It is easy to see that and, in particular, that r(m) ) ( ) m q(m) = () ) ( ) r() = q() It follows that it is easy to express the proaility of any event under the second hypothesis in terms of the function q. ote that the computation of the second factor is trivial y use of logarithms regardless of the size of. But equation () has another use. If a, and m are given from data, then the two hypotheses may e compared as estimated and we may ask: Which hypothesis is more proale given the data? We introduce, for fixed a and, the function φ defined y We have ) ( ) m φ(m) = r( ± µ) q() e µ / φ( ± µ) It is quite clear how q plays a fiduciary role here in the computation of proailities. However, given a, and φ, the value of m may e determined and, consequently, the value of µ. Thus, Bernoulli is ale to investigate deviations from. Example ( 3): In the previous paper, Bernoulli showed that when = 0000, q() = 0.00564 77. If now a =.055 (the oserved ratio of male to female irths), we find φ() = 0.000773 94 and so r() 940. Bernoulli, however, otains the slightly larger figure of 939 since he finds φ() = 96. Finally, Bernoulli asserts Pr(X 0, 000 a : = 055 : 000) = 469 0 939. The proaility 469 appears to e otained y multiplying r(0000) y 0. Implicitly, Bernoulli is saying that the tail proaility is no more than 0 times the oundary value. This is not a ad estimate. We have from the normal approximation with continuity correction Pr(X 0, 000 a : = 055 : 000) = 0.00007864 76. ow let m denote the value for which φ(m) =,, 3,.... Routine algera shows that we may write m = A + log φ log(a/) where A corresponds to the case φ =.
Example ( 6): If = 0000 and a : =.055, then A = 034 and we may write m = 034 + 43 log 0 φ It follows that if it were oserved that the numer of little oys orn were 034, then the evidence is equal for each of the two hypotheses. The special case, φ = 0 0 yields m = 034 + 430 = 0564. What is r(0564)? Bernoulli gives,46,000 which seems to have een applied here in error for it corresponds to using µ = 566. We have r(0564) =, 68, 377 q(0000) e 564 /0000 0 0, 56, 949 Bernoulli then asserts, without proof, that the proaility the numer of oys will exceed 0564 is 0 46000, whereas 0 46000 would have een more accurate. In fact, we have, with the usual continuity correction for the normal distriution, Pr(X > 0564 a : =.055) 74844. Example 3 ( 8): Bernoulli oserves that we may just as easily put r(m) q(m) = φ for which we will have m = A log φ log(a/) If all things are as efore, we find m = 034 43 log 0 φ = 034 430 = 9704. ow q(9704) q(0000) e 96 /0000 77 6384 =, 9, 968 Bernoulli has used µ = 300 so that e 300 /0000 8000 instead which yields for the proaility. Of course, r(9704) = q(9704) 0 0. By the normal approximation with continuity correction, Pr(X < 9704 a : = ) 7967. This then indicates that Pr(X < 9704 a : = ) 0 Pr(X = 9704 a : = ). If we now comine these results we find that 4600 Pr(9704 X 0564 a : =.055) 74844. Our next task is to find the maximum proaility of r given a and. Bernoulli oserves that we should have at the maximum r(m) = r(m + ). This leads to the solution M = a a + or, rather, 3 M = a a +
y oserving a and therefore the term in the numerator may e dropped. Of course, this is the standard result. Example 4 ( 6): If = 0000 and a/ =.055, we have M = a a+ = 068. Bernoulli now seeks a formula for the proaility of oserving any numer of oys out of irths. To this end he makes use of infinitesimal techniques. Let us put M = a a+, m = M + µ and π = r(m). We seek a formula for π when a and are nearly equal. ow Therefore r(m + ) = m m + a π = M µ M + µ + a π π = π M µ M + µ + a π corresponding to µ =. Passing to differentials, we have dπ [ π = M µ M + µ + a ] dµ This differential equation, suject to the initial condition Q = r(m) or rather, π = Q when µ = 0, has approximate solution π = Qe µ / as will now e demonstrated. The solution to the differential equation is ( ) [ + ln π = µ ) M + a ] ln(m + µ + ) + C suject to the initial π(0) = Q. This gives [( ) a + C = M + a ] ln(m + ) ln Q Consequently, we may write ln Q π = a + µ a + (M + ) ln M + µ + M + + ln M + µ +. M + as Bernoulli finds. To eliminate the factor of M + in the second term, we make use of the fact that so that ln M + µ + M + ln( + x) = x x + 3 x3 ( = ln + µ ) = µ M + M + ( ) µ + ( ) 3 µ M + 3 M + 4
truncated to 3 terms. Therefore ln Q π = a + ( µ M + µ 3 ) 3 (M + ) + ln M + µ + M + Further simplifications may e made. Since M is very large, we may replace M + µ y M. If further we assume µ M, then 3 (M+) may e ignored as well as M+µ+ M+. This then gives ln Q π = a + µ M Even more, if a is assumed nearly equal to, we have or ln Q π = µ M π = Qe µ /M Another approximation is otained as follows: Since M = a a+, a + µ M = a + µ (a + ) (a + ) = µ a 4a ut the factor (a+) 4a is very nearly if a is little different from. This then gives π = Qe µ / We return to the estimation of r(m) where M is the maximum proaility given y a and. Since M = a a a+, µ = M = a+ and q(m) = q()e µ / ) ( ) M r(m) = q(m) Bernoulli wishes to argue that r(m) q() or equivalently, that e a+ ) ) a ( ) a+ = To this end, put = and a = + α. We have ( ) e ( +α) α = ( + α) (+α) +α + α which, y taking logarithms, is ( ) α = α + + α + α ( ) ln ( + α) + ln + α 5
The Maclaurin series for each side agree up through order 3 and consequently Bernoulli concludes as long as α is very small, r(m) q(). By the previous paper, q() = 0.5643 As an aside we note that the numerator is in fact π (= 0.5643). We have q( ± µ) = 0.5643 e µ / Example 5 ( 7 8): Bernoulli first cites an example where of 5 irths, only 00 were oys. Under the assumption that a : =.055, the proaility that this occur is 4.7 0 66, somewhat larger than his estimate. In order to otain his result, we find that the numer of oys deviates from the hypothetical mean M = 65 y 63. Consequently, he estimates he estimates this proaility as.5643 e µ / =.5643 56 e 63 /56 = 3.3 0 70 In the same manner, if only 0 oys and 37 girls are orn, under the assumption that a : =.055, then the expected numer of oys is 9.6. The deviation of the numer of oys from this is 9.6. Therefore he estimates the proaility as.5643 e µ / =.5643 8.5 e 9.6 /8.5 = 0.005 9. In the previous paper, Bernoulli identified the quartiles of the inomial distriution with parameters = 0000 and p = as 0000 ± µ where µ = 47 4. Thus ( Pr 995 3 4 X 0047 p = ) 4 Actually, the proaility is 0.4960. What of a : =.055? As long as µ is small relative to and a/ the one distriution may e sustituted for the other changing only the location of the center. Thus, since we have M = 068, Pr (995 34 ) p + 68 X 00474 + 68 =.055 = or ( Pr 00 3 ) 4 X 035 p =.055 = 4 In reality, the proaility is 0.496. In general, we should oserve that the quartiles lie at µ = 0.475. and that the corresponding interquartile range is a a + ± 0.475 6
Thus, assuming as he does = 600000 offspring and a : =.055, we have a a + ± 0.475 600000 055 = ± 0.475 300000 = 30809 ± 59 055 Example 6 ( ): Bernoulli exhiits a tale of irths for the ten year period 7 730 in which these interquartile ranges are implicitly constructed. For example, in the year 7, = 8940 girls were orn and a = 9430 oys therey giving a total of = 8370. The quartiles lie at a distance of 0.475 985 = 45 from the mean. Under the hypothesis that a : =.055, the expected numer of oys orn is M = 943 which exceeds the oserved value y ut and so falls well within the interquartile range. For this reason, Bernoulli marks this case as B. Overall the ratio of oys to girls in this data is a : =.040. Under this hypothesis, the expected numer of oys orn is M = 9365 which falls short of the oserved numer y 65. 7