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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View Full Document(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 icemaking 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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 Spring '11
 Dr.Yun
 Probability theory, probability density function, Likelihood function, consultant

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