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hw1 - Homework Set No 1 Probability Theory(235A Fall 2009...

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Homework Set No. 1 – Probability Theory (235A), Fall 2009 Due: 10/6/09 1. (a) If (Ω , F , P ) is a probability space and A, B ∈ F are events such that P ( B ) 6 = 0, the conditional probability of A given B is denoted P ( A | B ) and defined by P ( A | B ) = P ( A B ) P ( B ) . Prove the total probability formula: if A, B 1 , B 2 , . . . , B k ∈ F such that Ω is the disjoint union of B 1 , . . . , B k and P ( B i ) 6 = 0 for 1 i k , then P ( A ) = k X i =1 P ( B i ) P ( A | B i ) . (TPF) (b) An urn initially contains one white ball and one black ball. At each step of the experiment, a ball is drawn at random from the urn, then put back and another ball of the same color is added. Prove that the number of white balls that are in the urn after N steps is a uniform random number in { 1 , 2 , . . . , N + 1 } . That is, the event that the number of white balls after step N is equal to k has probability 1 / ( N +1) for each 1 k N +1. (Note: The idea is to use (TPF), but there is no need to be too formal about constructing the relevant probability space - you can assume an intuitive notion of probabilities.) 2. If Ω = { 1 , 2 , 3 } , list all the possible σ -algebras of subsets of Ω. 3. Let (Ω , F ) be a measurable space. A pre-probability measure is a function P : F → [0 , 1] that satisfies P ( ) = 0 , P (Ω) = 1 . (P1)
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