By L.D. Meshalkin

The Russian model of a set of difficulties in chance concept features a bankruptcy dedicated to data. That bankruptcy has been passed over during this translation simply because, within the opinion of the editor, its content material deviates a bit of from that that is urged by means of the name: difficulties in seasoned bability concept. the unique Russian model comprises a few error; an test used to be made to right all blunders stumbled on, yet possibly a number of stiII stay. An index has been extra for the ease of the reader who can 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 booklet has been translated and edited with the desire of leaving as a lot "Russian taste" within the textual content and difficulties as attainable. Any pecu liarities current are probably as a result of the this purpose. August, 1972 Bryan A. Haworth viii Foreword to the Russian version This selection of difficulties in chance conception is basically meant for collage scholars in physics and arithmetic departments. Its aim is to aid the scholar of chance concept to grasp the idea extra professional foundly and to acquaint him with the appliance of chance idea the way to the answer of functional difficulties. This assortment is geared essentially to the 3rd variation of the GNEDENKO textbook direction in proba bility thought, Fizmatgiz, Moscow (1961), chance concept, Chelsea (1965).

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**Collection of problems in probability theory**

The Russian model of a suite of difficulties in chance concept 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 of from that that's prompt via the name: difficulties in seasoned bability thought.

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0063 if R is measured in parsecs. 4 Application of the formula for total probability 122. From among the 64 squares of a chess board, two different squares are chosen at random and two equal pieces of the white and black colors are placed on them. What is the probability that these pieces conquer one another if two rooks were placed? Two bishops? Two knights? Two queens? 123. From an urn containing 3 white and 2 black balls, there were transferred two balls, taken out at random, into an urn containing 4 white and 4 black balls.

The random variable I a) M[~J = ~ p{~ ~ are independent, positive and identically if k < n. takes on positive integer values. Prove that: m}; m~l b) D[~J = 2 I mP{~ ~ m} - M[~(M[~J + 1)]. m~l 165. Provethat D [~ 1 AJ = M [( ~ - M [~J2 1 A] - [M [~ 1 A] - [~MJY . 166. Assume that the random variable ~ coincides, with probability Pi' 41 Random variables and their properties with the random variable = D [~] ~i and let M[~;] =Mi • Prove that L Pk D [~kJ + D [flJ , k where fl takes on the values Mi with probability Pi.

Can pictures be glued on in such a way that events A and B are independent? 85. Suppose the random variables ~, and 11 are independent and identically distributed, with P{ ~ = I} =p>O, P{ ~ =O} = I-p>O. We introduce a new random variable, setting' = 0, if ~ + 11 is an even number and, = 1 if ~ + 11 is an odd number. For what values of p are the random variables ~ and, independent? I 86. 875. b) Prove thatP(A 2 1 Ad~ 1- P(A2 )fP(Ad. 87. Let P(A)=p, P(B)= 1-8, where 8 is small; estimate peA I B) from above and from below.