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

By William Feller

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There remain four intervals of length 116 each, and in each of them the graph of F differs from the whole graph only by a similarity transformation. Each of the four intervals therefore contains a subinterval of half its length in which F assumes a constant value (namely i, t, j, t, respectively). Continuing in like manner we find in n· steps 1 + 2 + 22 + ... + 2n-:-1 intervals of total length 2-1 + 2-2 + 2-3 + ... + 2- n = I - 2- n in each of which F assumes a constant value. 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.

1) which served to define Un. It suffices therefore to prove the theorem, f9r n = 2. But it is obvious that ({J2(t) I. 8). ~ = a 10. RANDOM DIRECTIONS Choosing a random direction in the plane j{2 is the same as choosing '. at random a point on the circle. 5: 8 < 217. For random' directions in the space j{3 the unit sphere serves as sample space; each domain has a probability equal to its 'area divided QY 417. Choosing a random direction in j{3 is equivalent to Readers who feel uneasy about the use of conditional probabilities in connection with densities should replace the hypothesis Xi == x by the hypothesis x - h < Xi < x, which has positive probability, and pass to the limit as h - o.

For. ) Am I ever going to be ahead of Mr. ~mith? dom walk interpretation the question is whether a first passage through + 1 will ever take place. As we know, this event has probability one, but the expected waiting time for it is infinite. Such waiting gives ample apportunity to bemoan my bad luck, and this only' grows more irritating by the fact that Mr. Smith argues iIi the same way. ~ 6. WAITING TIMES AND ORDER STATISTICS An ordered n-tuple (Xl" .. 6 This operation applied to all points of the space j\n induces n well-defined functions, which will be denoted by Xu>,"" X(n)' If probabilities are defined in 3{n these functions become random variables.

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