# 10 - April 2008 THE UNIVERSITY OF HONG KONG DEPARTMENT OF...

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Unformatted text preview: April 2008 THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT1302 PROBABILITY AND STATISTICS II Example Class 10 1. Let X 1 be a random variable having the pdf f ( x | λ ) = λ- 1 exp {- x/λ } , x > 0. Let x 1 be a realization of X 1 . (a) Show that P ( X 1 /λ ≤ u ) = 1- e- u ,u > . (b) Using (a), show how to find λ α ( X 1 ) such that P { λ α ( X 1 ) ≤ λ } = 1- α, for all λ > . How would u call the statistic λ α ( x 1 )? 2. At a busy telephone exchange, the times between the arrivals of the first 10 calls to arrive after 3:15pm one day are (in second) 4.10, 5.64, 1.16, 3.45, 0.91, 0.34, 0.44, 1.28, 0.64, 1.15, and the interarrival times of the first 10 calls to arrive after 6:15pm that same day are 0.37,0.69, 1.46, 0.19, 3.75, 0.64, 1.33, 1.65, 3.11, 0.95. Assume the data sets observed after 3:15pm and after 6:15pm constitute two independent exponential random samples with mean interarrival times θ 1 and θ 2 respectively....
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10 - April 2008 THE UNIVERSITY OF HONG KONG DEPARTMENT OF...

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