quiz_soln - ECE 534 September 16, 2009 Fall 2009...

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ECE 534 Fall 2009 September 16, 2009 Probability Quiz Solutions 1. (8 pts) Warm-up (a) Consider the probability space (Ω , F ,P ). If A ∈ F is such that P ( A ) = 1, then prove that for any C ∈ F , P ( A C ) = P ( C ). (Note: you cannot assume that A = Ω.) Ans: P ( A C ) = P ( A ) + P ( C ) - P ( A C ). But 1 P ( A C ) P ( A ) = 1. Thus P ( A C ) = 1, which implies that P ( A C ) = 1 + P ( C ) - 1 = P ( C ). (b) Determine if the following statement is True or False: Every (measurable) function of continuous-type random variable is also a continuous-type random variable. If you say “True”, explain why; if you say “False” provide a counterexample. Ans: False. We saw a counterexample in class. Hardlimiting a N (0 , 1) random variable to the interval [0 , 1] produces a mixed random variable with point masses at 0 and 1. (c) How many rolls of a fair die will it take on average to see a 3? Explain your reasoning carefully. Ans: It is easy to see that the number of rolls it will take to see a 3 is a Geom( ρ ) random variable with ρ = 1 / 6. Thus the average number of rolls is 6. Another way to see this it to realize that if m is the average, then m has to satisfy m = 1 + m (5 / 6), since if the first roll is not a 3, which happens with probability 5/6, the average from second roll onwards will be m as well. 2. (6 pts) Faulty Memory. A binary memory reader is faulty in a way that a bit ‘0’ is read as a ‘1’ with probability 0.2, and a bit ‘1’ is read as a ‘0’ with probability 0.1. Assume that the bits in the memory are equally likely to be ‘0’ or ‘1’. Also assume that the errors made by the reader are
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quiz_soln - ECE 534 September 16, 2009 Fall 2009...

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