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Midterm1 - Mathematics Department UCLA T Richthammer winter...

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Mathematics Department, UCLA T. Richthammer winter 09, midterm 1 Feb 02, 2009 Midterm 1: Math 170A Probability, Sec. 1 1. Axioms and properties of probability laws. (a) Carefully state the axioms of probability. (b) Use a Venn diagram to show P ( A B ) = P ( A ) + P ( B ) - P ( A B ). Answer: (a) The Axioms of probability are: Axiom 1: 0 P ( E ) 1 for all events E . Axiom 2: P ( S ) = 1. Axiom 3: P ( E 1 E 2 . . . ) = P ( E 1 ) + P ( E 2 ) + . . . for any sequence of disjoint events. (b) A Venn diagram show that A B = ( A - B ) ( A B ) ( B - A ) is a disjoint union, so by the addition property we get P ( A B ) = P ( A - B ) + P ( A B ) + P ( B - A ) and similarly P ( A ) = P ( A - B ) + P ( A B ), P ( B ) = P ( A B ) + P ( B - A ). Plugging this in shows that the assertion is true. 2. We choose two real numbers completely at random in the unit interval [0 , 1]. Describe a probabilistic model for this situation, and for the following events give a precise description and calculate their probability: (a) The first number is less than the second number. (b) The sum of the numbers is less than 3 / 2. (c) The product of the numbers is exactly 1 / 2. Answer: S = [0 , 1] 2 , where the projections X 1 , X 2 denote the coordinates of the first and second point. Let P be the equidistribution on S , i.e.
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