Elementary Applications of Probability Theory: With an by Henry C. Tuckwell

By Henry C. Tuckwell

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Recall that there are N - M type 2 individuals. If n ~ N - M all members of the sample can be type 2 so it is possible that there are zero type 1 individuals. However, if n > N- M, there must be some, and in fact at least n- (N- M), type 1 individuals in the sample. Thus the smallest possible value of X is the larger of 0 and n- N + M. Also, there can be no more than n individuals of type 1 if n ~ M and no more than M if M ~ n. Hence the largest possible value of X is the smaller of M and n. 1 Probability mass functions for hypergeometric distributions with various values of the parameters N, M and n.

The distance between them is then Z =IX- Yl. It is assumed that X and Y are independent and uniformly distributed on (0, 1). What is the probability density function of Z? 2 The density fz of Z is fz(z) = 2(1 - z), O~z~ 1. Proof The joint density of X and Y is given by f xr(X, Y) = 1, 0 ~x,y ~ 1. Refer to Fig. 4. 4 The unit square and the regions A1 and A 2 in which IX- YIE(z, z + dz). 20 Geometric probability We find Pr{ZE(z,z+dz)}= If fxr(x,y)dxdy A 1 uA 2 =2 Jx=z (fx-z l dy ) dx, y=x-(z+dz) the factor of 2 coming from symmetry considerations.

1. 2 . Find the density of Z =X+ Y. 11. Let U, V, W be independent random variables taking values in (0, oo). Show that the density of Y = U + V + W is fy(y)= J: J:fu(u)fv(z-u)fw(y-z)dudz. 12. With reference to the dropping of two points randomly in a circle, (i) Show P{r + drlat least one point is inS}= 2~r~P arccos(;,). 1). 1 ). 13. A point is chosen at random within a square of unit side. If U is the square of the distance from the point to the nearest corner of the square, show that the distribution function of U is nu, F u(u) = { 2u arcsin( 1 ;u2u) + J4u- 1, 1, (Persson, 1964).

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