� � 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 X1, . . . , Xn. 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 X1, . . . , Xn, where Xi is viewed as a Bernoulli random variable, with Xi = 1 if the ith 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 to construct a 95% confidence interval for θ, the proportion of people who support the candidate. As we saw in lecture, using the central limit theorem, an (approximate) 95% confidence interval can be defined as Θˆ− = Θˆn − 1.96 v v , Θˆ+ = Θˆn + 1.96 n n where v = Var(Xi), and Θˆn = (X1 + . . . + Xn)/n. Unfortunately, we don’t know the value for v. Construct confidence 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) n Sˆn 2 1 � (Xi − Θˆn)2 = n − 1 i=1 (b) Θˆn(1 − Θˆn) (c) The most conservative upper bound for the variance. f 1 Textbook problems are courtesy of Athena Scientific, and are used with permission.Page 1 oMIT OpenCourseWare http://ocw.mit.edu 6.041 / 6.431 Probabilistic Systems Analysis and Applied Probability Fall 2010 For information about citing these materials or our Terms of Use, visit:
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