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1. From among the students gathered for a lecture on probability theory one is chosen at random. Let the event A consist in that the chosen student is a young man, the event B in that he does not smoke, and the event C 3 Fundamental concepts in that he lives in the dormitory. a) Describe the event An BnC. b) Under what conditions will the identity An BAC= A hold? c) When will the relation CE B be valid? d) When does the equation A=B hold? Will it necessarily hold if all the young men smoke?159. The diameter of a circle is measured approximately. Assuming that its magnitude is uniformly distributed in the segment [a, b], find the distribution of the area of the circle, its mean value and dispersion. 160. The density of the distribution of the absolute value of the velocity183. The random variables & and 7 are independent and normally distrib uted, with the same parameters a and o. a) Find the correlation coefficient of the variables a$ + By and at - By ; also find their joint distribution. b) Prove that M [max ($, n)]=ato/ x.187*. Let { be an arbitrary random variable, where M=0, D[{]=02, and let F(x) be the distribution function of &. Prove that, for 02 x 0, F(x) > 2+ x Show by an example that these inequalities can turn into equalities for certain F's.231. Many botanists performed experiments on the crossing of yellow (hybrid) peas. According to a known hypothesis of Mendel, the prob- ability of the appearance of a yellow pea in such experiments equals $. In 34,153 crossing experiments, a yellow pea was obtained in 8506 cases. a) Assuming that the probability of obtaining a green pea in all experiments was constant and equal to 4, find the probability of the inequality 0.245