Photo web hosting - 3.4.1 NUMERICAL DISTRIBUTIONS 127 for some nonnegative integrable
3.4.1 NUMERICAL DISTRIBUTIONS 127 for some nonnegative integrable function g. If we set X c V/U, the probability that X 5 x can be calculated by integrating du dv over the region defined by the two relations in (26) plus the auxiliary condition v/u 5 x, then dividing by the same integral without this extra condition. Letting w = tu so that dv = udt, the integral becomes z dtm 15 udu=-2 J_ g(t) dt. s–cc I 0 co Hence the probability that X 5 x is /-~mS(tPt /~~+J7W. (27) The normal distribution comes out when g(t) = e-t2/2; and the condition u2 5 g(v/u) simplifies in this case to (w/u) 5 -4 lnu. It is easy to see that the set of all (u, v) satisfying this relation is entirely contained in the rectangle of Fig. 13. The bounds in steps R2 and R3 define interior and exterior regions with simpler boundary equations. The well-known inequality which holds for all real numbers x, can be used to show that l+lnc-cu 5 -1nu 5 l/cu-l+lnc (28) for any constant c > 0. Exercise 21 proves that c = e1j4 is the best possible constant to use in step R2. The situation is more complicated in step R3, and there doesn t seem to be a simple expression for the optimum c in that case, but computational experiments show that the best value for R3 is approximately e1.35. The approximating curves (28) are tangent to the true boundary when u = l/c. It is possible to obtain a faster method by partitioning the region into subregions, most of which can be handled more quickly. Of course, this means that auxiliary tables will be needed, as in Algorithms M and F. (5) Variations of the normal distribution. So far we have considered the normal distribution with mean zero and standard deviation one. If X has this distribu- tion, then y=p++x (2% has the normal distribution with mean p and standard deviation u. Furthermore, if Xl and Xz are independent normal deviates with mean zero and standard deviation one, and if K =Pl+alXl, 35 = P2 + c72(PXl + &=-7X2), (30) then Yi and Y2 are dependent random variables, normally distributed with means ~1, ~2 and standard deviations ui, us, and with correlation coefficient p. (For a generalization to rz variables, see exercise 13.)