# ESM3A HW6 - Jacobs University Bremen School of Engineering...

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Jacobs University Bremen School of Engineering and Science Peter Oswald Fall Term 2010 120201 ESM3A — Problem Set 6 Issued: 13.10.2010 Due: Wednesday 20.10.2010 (in class) This is about basic properties of probability spaces, conditioning and independence of events. Compare e.g. Lectures 8 and 9 in the posted script. 6.1. (12 pts) Suppose we have a probability space (Ω , Σ , P ), i.e., a sample space Ω, a collection Σ of its subsets (called events of interest), and a probability function P : Σ [0 , 1] (also called probability measure) that satisfy the properties mentioned in class. a) Suppose, we define new functions Q ( A ) = [ P ( A )] 2 resp. R ( A ) = P ( A ) / 2 for A Σ. Is , Σ , Q ) a probability space? Why or why not? Same question for (Ω , Σ , R ). b) Let A 1 , A 2 , A 3 , . . . be a sequence of events in Σ. Prove that P ( S n =1 A n ) 5 n =1 P ( A n ). c) If you pick a real number from [0 , 1] at random than the probability that you pick a particular number, should be zero (why?). What is the probability that you pick a rational number? Hint : Write down which events are involved, and think how many rational numbers we have in [0 , 1]. Part b) or the countable additivity property of P may be used here. 6.2. (8 pts) a) A committee of 5 is formed by election from a list of candidates including 6 men and 5 women. Suppose the candidates are equally qualified and liked. What is the

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