Read e-book online An Introduction to Probability Theory and Its Applications, PDF

By William Feller

ISBN-10: 0471257095

ISBN-13: 9780471257097

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Extra resources for An Introduction to Probability Theory and Its Applications, Vol. 2

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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.

1) combined with the simplicity of the proof are apt to arouse suspicion. 1) from the. direct definition of the probability in question as the (n+ 1)-tuple integral of ocn+le-cz(xo+"-+:rn) over the region defined by the inequalities 0 < Xo < Xn and 0 < ~i < Xo for I = I, . . , n-1. 1) is an instructive exercise in conditional probabilities; it is less simple, but leads to additional results (problem 8). Given that Xo = x, the probability of a greater value at later trials is p = e- czx , and we are concerned with the waiting time for the first "success" in Bernoulli trials with probability p.

This convenient and intuitive device has been used since the beginning of probability theory, but it depends on neglecting events of zero probability. (d) Cantor-type distributions. 4) Y = 3~4-TXv. v-I (The factor 3 is introduced to sjmplify the discussion. ) The distribution function F(x) = P{Y < x} will serve as example for so-caHed singular distributions. In the calcllation we refer to Y as the gain of a gambler who receives the amount 3· 4-k if the kth toss of a fair coin results in tails. This gain lies between 0 and 3(4-1 +4- 2 + ...

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An Introduction to Probability Theory and Its Applications, Vol. 2 by William Feller

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