18.440, March 9, Stirling s formula

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1 Stirlig s formula 8.44, March 9, 9 The factorial fuctio! is importat i evaluatig biomial, hypergeometric, ad other probabilities. If is ot too large,! ca be computed directly, by calculators or computers. For larger, usig there are difficulties with overflow, as for example 7! >, 54! > 5, which overflows o oe calculator I have, which computes 53!. Also, direct multiplicatio of may factors becomes iefficiet. There is a relatio with the gamma fuctio,! Γ+), where Γα) + x α e x dx. The statistical computig system R i the versio we have as of this date) ca fid 7! Γ7) but it balks at Γ7), so it breaks dow for smaller tha the calculator does. Of course, some computer systems ca fid! for very large. Mathematica gave! exactly, showig all the may digits, which is ot ecessarily coveiet. Stirlig s formula provides a approximatio to! which is relatively easy to compute ad is sufficiet for most purposes. Usig it, oe ca evaluate log! to better ad better accuracy as becomes large, provided that oe ca evaluate log as accurately as eeded. The to compute bk,, p) : k) p k q k, for example, where < p q <, oe ca fid log bk,, p) log! log k! log k)! + k log p + k) log q. The probability bk,, p) caot overflow, ad i iterestig cases it will also ot uderflow /bk,, p) will ot overflow). Two sequeces of umbers, a ad b, are said to be asymptotic, writte a b, if lim a /b. This does ot imply that lim a b ) : for example, + but + ) teds to with. But a /b is equivalet to log a ) log b ) log a /b ). Theorem. Stirlig s formula.! π +/) e e π. Thus, [ log!) + ) log + ] log π) as. Proof. The sig : will mea equals by defiitio. Let d : log!) + ) log +. The we eed to prove d coverges to a costat, [log π)]/. First, d d + log + ) + ) log ) log + ) + ) ) + log. We have the Taylor series log + t) t t + t3 3

2 for t <. For t > the terms alterate i sig. A trasformatio will help to get terms of the same sig. The trick is to otice that The log ) + t t33 t55 log + t) log t) t ), t where ow all terms are of the same sig. Thus ) d d + + log ) ) ) ) 6 + >. So d decreases as decreases. Comparig the last series to a geometric oe with ratio + ) gives < d d + < + ) 3[ + ) ] 3[ + ) ] + ) + ), so d < d + + ). So we see that d /) icreases as does. As, d decreases to some C with C < + ad d /) icreases up to some D with < D +. Sice /) coverges to, we must have < C D < +, ad d coverges to a fiite limit C. By defiitio of d we the have!/ +/ )e e C or! e C +/ e. The last step i the proof is to show that e C π) /. This will ivolve aother famous fact: Theorem. Wallis product. π 7 m m m m +, or π lim 4m m!) 4 m m)!m + )!. Remarks. To see the relatioship betwee the two statemets, first ote that m ) ) 3) m) m m!, the that m + ) m + )!/ 4 6 m), etc. Note that the product coverges to π/ rather slowly; it would ot give a good way to compute π.

3 Proof. Itegratig by parts gives, for, si xdx si xdcosx) cos x si x + ) si x cos xdx cos x si x + ) si x si x)dx, so si xdx cos x si x + ) si xdx, ad si xdx cos x si x + si xdx. Thus ) si xdx si xdx. The for m,,..., iteratig ) gives si m xdx m m m 3 m π sice si m+ xdx m m + m m 3 sice dx π. si xdx. Let A m : si m xdx/ si m+ xdx. The π A m 7 m m Now we will prove lim m A m. For x π/, < Now by ) above, si m+ x si m x si m x, so si m+ xdx < si m xdx < m for all m,,.... m + si m xdx. si m+ xdx/ si m xdx m m + as m ad si m xdx, beig betwee umerator ad deomiator, also has the ratio A m covergig to, provig Wallis product. 3

4 Now to fiish provig Stirlig s formula, let B : e C. As,!e / +/ B, )!e /) +/ B, ad!) e / + B. Dividig gives!) +/ /[)! / ] B. Now, Wallis product gives!) /[)! + ) / ] π/) /. Sice + ) / ) /, we get b/ / / π/) /, B π) /, provig Stirlig s formula. The proof provides further iformatio o how good a approximatio Stirlig s formula gives to!. Sice d > C > d /), where C [log π)]/, so C < d < C + /), we have the bouds 3) π) / +/ e <! < π) / +/ e +[/)]. Eve closer bouds are available. From ), d d + j 3 j + ) > j 5 ) 9 + ), so 4 d d + > > 3 + ) 3 + ) ) ) ) ) + ) [ ] ) 8 + ) [ ] ) 8 + ) [ + [ ) 4 + ) ] + ) 3 + ) + > + ) ) sice ) ) ) ) 3. So, d > d + + ) ), 3 ad the sequece d /) + /36 3 ) decreases as dow to its limit, which is also C, so d /) + /36 3 ) > C. Writig expx) : e x, we have the followig improvemet o the left side of 3): for all,,..., 4) π +/ exp + ) 36 3 <! < π +/ exp + ) As, the ratio of the upper to lower boud coverges to rather fast sice /36 3 ) rather fast. 4 ].

5 There are further improvemets, although they wo t be proved here: Whittaker ad Watso, Moder Aalysis, p. 5, gives a asymptotic expasio d C The series does ot coverge for ay, but if the sum of the first k terms is used as a approximatio to the left side d C, the error i the approximatio has the same sig as, ad smaller absolute value tha, the ext k + )st) term. This was proved above for k by 3) ad for k by 4). Now Stirlig s formula with error bouds ca be used to give upper ad lower bouds for ) k : ) k + )!/ k)! for itegers k. Specifically, 4) implies +/ exp + ) k < k) k+/ exp +/ exp + ) k > k) k+/ exp +/ k ) + 36 k) 3 k) + k) ) 36 k) 3 k) + k) ) ad ). ). The above iequalities o ) k Let j, k) : exp k) k+/ k) show that it is approached by j, k) withi a factor of exp[/36 k) 3 )], which is very close to if k is large. For k large, exp[ k/ k))] also approaches, although ot as fast. Let p, k) : ) k / k, the probability that k umbers, chose at radom from,..., with replacemet, are all differet. The, to the accuracy of the above approximatio for ) k, p, k) is approximated by e k k ) k+/ ) k exp. k) For a simpler ad rougher approximatio, omit the exp... factor. Now, suppose that for a give ad α, with < α <, we wat to fid the smallest k such that p, k) < α. For example, if 365 ad α /, the questio is how may people are eeded to give a eve chace that at least two of them have the same birthday eglectig leap years ad assumig that births are evely distributed throughout the year). To fid the desired k, oe ca compute p, k) ad use trial ad error. To speed up the process oe ca use a simpler approximatio where we ca solve for k to get a good first approximatio to k. The most likely oly a few values of k ear the first oe eed to be tried. Here is how oe ca get such a simple approximatio. For k <, the Taylor series of log x) gives log k ) k k k

6 If k/ is small, later terms i the series ca be eglected, ad log p, k) is approximated by 5) log p, k) k k + ) k k3 6 + k +, where the ext largest terms would be of the order of k 4 / 3 ad k / ad k/[ k)] is still smaller). Note that if we approximated log k/) bu just the first term k/, we would ot eve get the first term i 5) correct the i the deomiator would be missig). Usig the first term k /) i 5) as our first approximatio, solvig for k gives k /) log α, or 6) k [ log α)] /. For such a k, the ext two terms i the approximatio are smaller by factors of the order of / /, so they ca be reasoably be eglected if is large. This gives a Method. To fid the least k such that p, k) < α, for give ad α, first try k as the ext larger iteger tha the umber from 6). Compute p, k). If p, k) < α, check that p, k ) α. If ot, cosider k, etc. util a solutio is foud. If p, k) > α, fid whether p, k + ) < α. If so, the solutio is k +. If ot, try k +,..., util a solutio is foud. Example. The birthday problem. Here 365 ad α /. First try k as the ext iteger larger tha log) /, that is k 3. The we fid p365, 3) < /, so we ext compute p365, ) ad fid it is larger tha /, so k 3 is the solutio: i a group of 3 or more people, there is a better tha eve chace that at least two have the same birthday. Example. A computer pseudo-radom umber geerator starts with a umber s called a seed ad uses a fuctio f to geerate umbers s s, s fs ), s 3 fs ),..., s j+ fs j ), l.... Suppose that the umbers s j will be itegers from a to for some, ad f is a radomly chose fuctio from the set {,,..., } ito itself, where each of the such fuctios is equally likely. For how large r will there be a eve chace that s r s m for some m < r? Oce this happes, the s r+ s m+, etc. ad the s i will go roud ad roud a closed cycle. So the evet that s r s m for some m < r is the evet that the s j for j r are ot all differet. The above method applies with α /. If 6, for example, 6) gives r 78 ad it ca be checked that p, 78) < / < p, 77). So i this case there is a eve chace that the geerator will fall ito a closed cycle after oly 78 of the,, available umbers. By the way, the average legth of the closed cycle is just half of the first umber r such that s r s m for some m < r. So there is a paradox: a truly radom fuctio f makes a bad pseudo-radom umber geerator. Better geerators are made by usig umber-theoretic methods to assure that there are o short closed cycles. Bibliographic Notes. James Stirlig published his formula i Methodus Differetialis 73). Abraham De Moivre, aother mathematicia ad fried of Stirlig s, discovered 6

7 the formula except for fidig the value of the costat factor π) /. The proof of the formula ad up through 3) above is due to Herbert Robbis, Amer. Math. Mothly 6 955) pp The refiemet of the proof to give 4) is due to T. S. Najudiah, ibid ) pp As metioed, further terms i the asymptotic expasio ext display after 4)) ca be foud from E. T. Whittaker ad G. N. Watso, Moder Aalysis Cambridge Uiv. Press, 4th ed., 97, repr. 96) pp Joh Wallis published his product without a real proof) aroud 65 see his Opera Omis, re-published i 97). The above proof came from R. Courat, Differetial ad Itegral Calculus I, d. ed., traslated by E. J. McShae Itersciece, N. Y., 937). Stirlig s formula examples Let S) /e) π) /. The! S) as, meaig!/s), ad!/[s)e /) ] faster. But! S) does ot coverge to ; i fact it icreases very fast, but ot as fast as! or S).! S)! S)!/S)!/[S)e /) ] Note that the ratios i the ext to last colum decrease toward. They are approximately + /). The ratios i the last colum icrease toward, faster. They are approximately /36 3 ). So as becomes large, i terms of ratio ot differece),! is fairly well approximated by S), much better approximated by S)e /), ad still much better approximated by S) exp[/) /36 3 )]. For large, oe eeds to take accout of roudig error. I log +.5 ) +.5) log, a roudig error i log is multiplied by. If is k, for example, this meas a loss of k decimal places of accuracy. 7

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