By Henry C. Tuckwell

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**Extra resources for Elementary Applications of Probability Theory: With an introduction to stochastic differential equations**

**Example text**

3) Thirdly, consider Fig. 7. Given at least one point is inS, the probability that the distance separating the points is between p and p + dp is the area of S' divided by the area of C. It is shown in the exercises (Exercise 12) that this is P{r+dr[atleastonepointisinS } = 2~r~p arccos(;r). 3): Pr {at least point is inS}= 4 dr + o(dr). 2) gives ( p )4dr 4dr) + o(dr). - + -2pdp P{r + dr} = P{r} ( 1 2-arccos 2r r rcr r Then (pr )] dr + o(dr). 8pdp -4P +~arccos dP = P{r + dr}- P{r} = [ -r2 Multiplying by r 4 and rearranging gives 8pdpr arccos ( p) r 4 dP + 4r 3 P dr = -rc2r dr + o( dr).

96 in this formula. Discussion The above estimates have been obtained for direct sampling in the ideal situation. Before applying them in any real situation an examination of the assumptions made would be worth while. Among these are: (i) The marked individuals disperse randomly and homogeneously throughout the population. (ii) All marked individuals retain their marks. (iii) Each individual, whether marked or not, has the same chance of being in the recaptured sample. (iv) There are no losses due to death or emigration and no gains due to birth or immigration.

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