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By L.D. Meshalkin

The Russian model of a set of difficulties in likelihood conception includes a bankruptcy dedicated to facts. That bankruptcy has been passed over during this translation simply because, within the opinion of the editor, its content material deviates a bit from that that is urged via the identify: difficulties in professional­ bability idea. the unique Russian model includes a few error; an try out was once made to right all blunders chanced on, yet might be a number of stiII stay. An index has been further for the ease of the reader who might be trying to find a definition, a classical challenge, or no matter what. The index lists pages in addition to difficulties the place the listed phrases seem. The e-book has been translated and edited with the wish of leaving as a lot "Russian taste" within the textual content and difficulties as attainable. Any pecu­ liarities current are probably end result of the this purpose. August, 1972 Bryan A. Haworth viii Foreword to the Russian variation This number of difficulties in chance conception is essentially meant for collage scholars in physics and arithmetic departments. Its objective is to assist the scholar of likelihood idea to grasp the idea extra professional­ foundly and to acquaint him with the applying of chance concept the right way to the answer of sensible difficulties. This assortment is geared essentially to the 3rd variation of the GNEDENKO textbook path in proba­ bility concept, Fizmatgiz, Moscow (1961), likelihood concept, Chelsea (1965).

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The tickets are taken out randomly one at a time (without returning). What is the probability that: a) for at least one drawing, the number of the selected ticket coincides with the number of the trial being performed; b) If m tickets are drawn (m

The random variables ~ and 11 are independent, where M[~J =2, M[rJJ=I, D[rJJ=4. Find the expectation and dispersion of: a) (1 =~-211; b) (2=2~-11. D[~J=I, 151. Assume that in a lake there were 15,000 fish, 1000 of them marked [with radioactive tracers] (see Problem 100). 150 fish were fished out of the lake. Find the expectation of the number of marked fish among the fish caught. 152. Find the expectation and the dispersion of the number of short fibers among the randomly selected fibers in Problem 94.

175. A certain number of perfectly spherical balls, made from a homogeneous material, yield a symmetric distribution when grouped according to diameter. , the third central moment will be positive). 176. Prove that an arbitrary distribution function possesses the following properties: f~ lim x X-I'oo Z f~ 00 00 dF (z) = 0, lim x x--+o+ X Z dF (z) = 0. x 177. Prove that if a random variable ~ has a moment of order k, then limx--+ooxk(I-F(x)+F( -x))=O. 178. , for an arbitrary m and arbitrary real Xl' X2, ...

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