By William Feller
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Additional resources for An Introduction to Probability Theory and Its Applications, Vol. 2
With trite verbal changes this model applies also to splittings of. mineral grains or pebbles, ~tc. Instead of masses one considers' also energy losses under collisions, ,and the descript~on simplifies somewhat if one is concerned with changes of energy of the same particle in successive collisions. As, a last example consider the changes the intensity of light when passing through' matter. Ex~mple 10(a) shows that when a light ray passes through a sphere of radius R "in a random direction'" the distance traveled througp, the sphere is distributed 'uniformly between and 4R.
Th us F is a continuous function increasing from F(O) = 0 to E(l) = 1 in such a 11'ay that the intervals of constancy add up to length 1. Roughly speaking: the whole increase of F takes place on a set of measure O. We have here a continuous distribution function F without density f ~ 12. EMPIRICAL DISTRIBUTIONS The" empirical distribution function" Fn of n points a1 , • •• ,an on the line is the step function with jumps l/n at 1 , ••• , an' In other words, . a I n Fn(x) equals the number of points ak in - 00, x, and Fn is a distribution function.
The reason is that only rationals can have two expansions, and the set of all rationals has probability zero. t: ... 11 THE USE OF LEBESGUE MEASURE 35 (c) Coin tossing and random choice. Let us now see how a "random choice of a point X between 0 and 1" can be described in tenns of discrete random variables. Denote by Xk(x) the kth decimal of x. ) The random variable Xk assumes the values 0, 1, ... , 9, each with probability T\-' and the Xk are mutually independent. 2) This formula reduces the random choice of a point X to successive choices of its decimals.
An Introduction to Probability Theory and Its Applications, Vol. 2 by William Feller