Finding the Distribution of a Random Variable from its Moment Function
Otwombe, N. K.
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Consider the problem of r .~/ randomly distributed points in a unit n-ball and the convex hull created by these points. Let ~II be r! times the r-content of an rsimplex whose P vertices are in the interior and r -I- /- p vertices on the boundary of a unit n-ball. Explicit expressions for the exact distribution functions of ~II are given when r I / points are independently, and identically distributed according to the Uniform distribution. The exact distributions are obtained using the technique of Inverse Mellin transforms with the help of the moment functions. The technique is illustrated for the general case p =- r -j J and a particular case p =3, r - 2 . Various representations of the distributions in psi and the generalized zeta functions are given. These representations are also given in the most general case as an H- function distribution. SYMBOLS AND NOTATIONS The following is a list of symbols and notations, with meanings indicated on the right that will frequently occur in this Research. (" ) = 17 1 fII ml(n-m) Binomial Coefficient m-J .(at, =n(a +.j), (a),. = I Pochhammer Symbol j (! r(a) Gamma Function pdf Probability density function The cumulative distribution function The natural logarithm of 10 r-content of the r-simplex generated by r + 1 points 11" = r! 11 r! times the r - content of the simplex R",£" Euclidian n-space Re(.) The real part of (.) arg(.) Argument of (.) GRAPHS AND TABLES Fig. 4.1 Theoretical cdf plot for r=2, n=Z, p=3 Fig. 4.2 Density plot for r=2, n=Z, p=3 Table 1 Table of Theoretical moments and the exact Moments from Equation (4.1.1)