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# Assignment2 - 2/2011 THE UNIVERSITY OF HONG KONG DEPARTMENT...

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2/2011 THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT1302 PROBABILITY AND STATISTICS II Assignment 2 1. For each of the following situations, (i) identify the observed data , (ii) specify the unknown parameter and its corresponding parameter space , and (iii) write down the likelihood function based on the data. (a) Take 1,000 ﬁsh from a lake containing N ﬁsh, mark them, and return them. Over the next month, anglers catch 2,000 ﬁsh, 100 of which are marked. (b) A coin has a probability p to turn up a head when tossed. It was tossed 100 times and 60 heads turned up. (c) Assume that the interarrival times of customers to a service centre are i.i.d. exponential random variables with constant rate θ > 0. During the half hour from 4:00pm to 4:30pm, customers arrived at times 4:08pm, 4:12pm, 4:15pm, 4:23pm and 4:29pm. (d) A group of 108 patients suﬀering from a certain illness is divided into two subgroups of the same size; the ﬁrst subgroup receives ordinary treatment, the remainder receives a new treatment. Suppose the probabilities of substantial, mild and no improvements for the two treatments are as follows: Ordinary treatment New treatment Substantially improved p s q s Mildly improved p m q m Not improved p n = 1 - p s - p m q n = 1 - q s - q m . The following results were observed: Ordinary treatment New treatment Substantially improved 12 18 Mildly improved 6 12 Not improved 36 24 Total 54 54 (e) Assume the number of bus breakdowns on a single day is a Poisson ( λ ) random variable. A bus company experienced three bus breakdowns yesterday. 1

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(f) Assume the height of a student is normally distributed, with mean μ 1 , variance σ 2 1 for boys, and mean μ 2 , variance σ 2 2 for girls. The heights of 9 randomly selected students were measured to be 1.72, 1.76, 1.80, 1.65 m for males, and 1.56, 1.48, 1.62, 1.55, 1.58 m for females. (g) A sample of size 5 is drawn at random without replacement from a population of size N = 2000, in which there are m males and N - m females and m is not known. It is found that there is only one male in the selected sample. (h) An ice-making machine produces ice cubes of nominal volume 1 cm 3 . Assume the actual volume of an ice cube produced is normally distributed with mean 1 cm 3 and variance σ 2 . In a production run, volumes of 8 ice cubes were measured as follows (in cm 3 ): 0.82, 1.01, 1.03, 0.92, 0.84, 1.10, 0.78, 0.93. A random sample of size 100 was found to consist of 3 red, 24 yellow, 30 brown and 43 (j) Flaws appear at random on sheets of a certain type of photographic paper, so that the number of ﬂaws on a single sheet follows a Poisson ( λ ) distribution. On 6 separate sample sheets, 12, 2, 4, 6, 5 and 10 ﬂaws were detected respectively. 2.
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Assignment2 - 2/2011 THE UNIVERSITY OF HONG KONG DEPARTMENT...

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