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# Hw3 - EE 178 Probabilistic Systems Analysis Homework#3 Due...

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EE 178 Handout #4 Probabilistic Systems Analysis January 24, 2008 Homework #3 Due Thursday, January 31 1. Liars. Of the 100 people in a village, 50 always tell the truth, 30 always lie, and 20 always refuse to answer. A sample of 30 people are drawn with replacement. a. What is the probability that the sample will contain 10 people from each category? b. What is the probability that the sample will have exactly 12 liars? 2. Binary vectors. The sample space Ω of an experiment consists of all 8-dimensional binary vectors, e.g., every member of Ω has the form ω = ( ω 0 , . . . , ω 7 ), where ω i is 0 or 1. The probability law P assigns a probability of 1 / 2 8 to each of the 2 8 elements in Ω. Find the pmfs describing the following random variables: a. W ( ω ) = 7 i =0 ω i , i.e., the number of 1’s in the binary vector. b. X ( ω ) = 1 if there are an even number of 1’s in ω and 0 otherwise. c. Y ( ω ) = ω 4 , i.e., the value of the 4th coordinate of ω . d. Z ( ω ) = max i ( ω i ). 3. Heads and tails. Consider a sequence of five independent fair coin tosses. Define random variable X to be the number of times that heads is followed immediately by a tail. Find

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