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rec23 - 1200 people in North Carolina and found that 684...

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Massachusetts Institute of Technology Department of Electrical Engineering & Computer Science 6.041/6.431: Probabilistic Systems Analysis (Fall 2010) Recitation 23 December 2, 2010 1. Example 9.1, page 463 in textbook Romeo and Juliet start dating, but Juliet will be late on any date by a random amount X, uniformly distributed over the interval [0 , θ ]. The parameter θ is unknown. Assuming that Juliet was late by an amount x on their first date, find the ML estimate of θ based on the observation X = x . 2. Example 9.4, page 464 in textbook Estimate the mean μ and variance v of a normal distribution using n independent observations X 1 , . . . , X n . 3. Example 9.8, page 474 of textbook We would like to estimate the fraction of voters supporting a particular candidate for office. We collect n independent sample voter responses X 1 , . . . , X n , where X i is viewed as a Bernoulli random variable, with X i = 1 if the i th voter supports the candidate. We conducted a poll of 1200 people in North Carolina, and found that 684 were supporting the candidate. We would like
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Unformatted text preview: 1200 people in North Carolina, and found that 684 were supporting the candidate. We would like to construct a 95% conFdence interval for θ , the proportion of people who support the candidate. As we saw in lecture, using the central limit theorem, an (approximate) 95% conFdence interval can be deFned as ˆ Θ-= ˆ Θ n-1 . 96 r v n , ˆ Θ + = ˆ Θ n + 1 . 96 r v n where v = Var( X i ), and ˆ Θ n = ( X 1 + . . . + X n ) /n . Unfortunately, we don’t know the value for v . Construct conFdence intervals for θ using the following three ways of estimating or bounding the value for v (in each case simply assume that v is equal to the given estimate; note that this is a further approximation in cases (a) and (b)). (a) ˆ S 2 n = 1 n-1 n s i =1 ( X i-ˆ Θ n ) 2 (b) ˆ Θ n (1-ˆ Θ n ) (c) The most conservative upper bound for the variance. Page 1 of 1...
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