192 



FISHERY BULLETIN OF THE FISH AND WILDLIFE SERVICE 



the number of marked individuals in a sample of 

 size 71." Laplace treated this problem as an urn 

 problem, with an infinite number of black and 

 white balls representing marked and umnarked 

 members. On the basis of an extension of Bayes' 

 theorem, he predicted from a first sample of t 

 and n observed what a second sample with known 

 T but unknown A^^ might produce. He found 

 that the mean value of N would be equal to 



—7- if T, n, and / are all large. He also took the 



Tn 

 distribution of A'^ to be normal about — as mean 



with standard deviation estimated by 



where t, u, and T are all large, Laplace's case, 



Cn 



Tu{T+t){t+u) 



(2) 



where the numbers are all large. 



For the preceding example, where T=]00, 

 71 = 500, and t=50, Laplace's solution would give 

 an estimate of standard deviation 



/I 



00X450X1 50 X500 Y^^g^ 



50* 



Pearson reexamined this problem in his 1928 

 paper because he felt Laplace's urn statement did 

 not fit the actual problem since "We are not 

 taking a second sample from an infinite population. 

 We have only one sample and we want to learn 

 something about the population from which it 

 has been sampled, which is finite in extent, 

 although its extent is unknown. We do know, 

 however, that it contains T white balls; i. e., 

 births in all France." 



Assuming the sample ti to be a random sample 

 of the finite population A'^, and on the basis of 

 inverse probabilities (Bayes' theorem), Pearson 

 finds that the modal value of the distribution of 

 the possible values of A'^ is 



N=u + T- 



u{T-t)_nT 



t 



the mean value is 



A^=ii+T- 

 and the variance is 



{t-2) 



On 



2 



(•u+l)(r-<+l) (n- 1) (T-1) 



{t-2y {t-2.) 



(3) 



(4) 



(5) 



Ar=7V=f 



and 



"N 



TuiT-t)(t+u) 

 t^ 



(6) 



(7) 



This estimate of o-^^ is different from and smaller 

 than that of Laplace, the disagreement being 

 attributed by Pearson to Laplace's taking his 

 sampled population as if it were a second sample 

 independent of that already taken. 



For the example employed before, with r=100, 

 71=500, <=50, formula 7 would give 



an 



<' 



00X450X50X500X1 



50' 



=9; 



Pearson's paper seems to have been generally 

 overlooked by zoologists dealing with similar 

 problems. 



SOME APPLICATIONS IN THE LITERATURE 



Formula 1 has been applied to the estimation of 

 diverse aniinal populations. One of the best 

 known of these applications is the so-called Lincoln 

 index of the duck population of North America de- 

 veloped by Lincoln (1930), and mentioned in the 

 textbook of Leopold (1935), the monograph of 

 Kendeigh (1944), the manual of Wright (1939), 

 and elsewhere. Lincoln used the ducks banded at 

 stations in North America as his marked members, 

 and the kill by hunters as his sample of the popula- 

 tion. The inaccm-acies of kill records and the in- 

 complete return of bands were recognized as 

 sources of errors. No attempt was made to esti- 

 mate the statistical error. 



An application of tliis method was made by Vor- 

 hies and Taylor (1933). These workers computed 

 the number of jack rabbits on fenced cattle ranges 

 of Arizona by taking the ratio of jack rabbits seen 

 to the number of cattle seen in a strip of width 

 equal to the apparent flushing distance of the 

 jack rabbits, and comparing this ratio with the 

 known number of cattle on the range. In this 

 case, the cattle would represent the "marked" 

 members of the population of rabbits plus cattle. 

 It seems rather doubtful whether the ratio in the 

 sample would be a fair estimate of the ratio in 

 the population because of the obviously different 



