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Unformatted text preview: THE UNIVERSITY OF HONG KONG
DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE Pmbayuwj Md mrbﬂ‘c: I
STAT1301 S—I—‘ATTSTTGSjﬁrN'D—PRO‘BKBEITY‘I December 28, 2005 Time: 2:30 p.m. — 4:30 p.m. Candidates taking examinations that permit the use of calculators may use any cal—
culator which fulﬁls the following criteria: (a) it should be selfcontained, silent,
battery—operated and pocketsized and (b) it should have numeral—display facilities
only and should be used only for the purposes of calculation. It is the candidate ’3 responsibility to ensure that the calculator operates satisfactorily
and the candidate must record the name and type of the calculator on the front page
of the examination scripts. Lists of permitted/prohibited calculators will not be made
available to candidates for reference, and the onus will be on the candidate to ensure
that the calculator used will not be in violation of the criteria listed above. Answer ALL FIVE questions. Marks are shown in square brackets. 1. Consider the random experiment of ﬂipping an unfair coin four times. Assume
that at each trial (flip), the probability that the head appears is 2 / 3, the prob
ability that the tail appears is 1/3 and that different trials are independent. Let A and B be two events deﬁned as follows: A = {At least one tail appears},
B 2 {At least three heads appear}. (a) Find the conditional probabilities P(AB) and P(BA). [6 marks] (b) Are A and B independent? Give reasons for your answer. [4 marks] Total: [10 marks] 2. (a) Let X and Y be two independent Poisson random variables with pa
rameters A1 = 2 and A2 = 3, respectively (i.e. X N Poisson (2),
Y N Poisson (3), and X and Y are independent). (1) Find the probabilities P{X > 3} and P{X = 1, 1 < Y s 3}. [4 marks]
(ii) Let z = XY. Obtain P{Z = 0}, P{Z = 3} and P{Z : 4}. [7 marks] S&AS: STAT1301 Statistics and Probability I 2 (iii) Find E (Z) and Var(Z) where Z is deﬁned in (ii). [5 marks]
(b) Suppose U ~ N(0, 1) and V N N(—2, 12.25) and that U and V are
independent.
(i) Find the probabilities P{—2 S V S 3.5}, and
P{U > 1, 1.5 < V g 5}. [4 marks] (ii) Let W = 5U + 4V + UV. Find E(W) and Var(W).
[Hint You may use the following formula directly (a + b + c)2 = a2 + b2 + c2 + 2ab + 2ac + 2bc] [6 marks]
Total: [26 marks] 3. (a) Let the joint probability mass function of two discrete random variables
X and Y be given by x+y
32 ’ p($ay): 56:1)21 y=172a3)4'
(i) Find the marginal probability mass functions p x (as) and py (y) of X
and Y, respectively. [4 marks] (ii) State, with reasons, whether X and Y are independent. [4 marks]
(iii) Find the probabilities P{X > Y} and P{X + Y = 3}. [4 marks] (b) In the early part of the twentieth century, newspaper distributors in many
towns in Britain used to hire boys to sell newspapers. Assume the fol
lowing scenario: A newspaper boy would buy his papers for 25 pence
each and would sell them for 35 pence each. Unsold papers could be
given back to the distributor and the boy could receive 20 pence for each
returned paper. Suppose that the number of newspapers required in the
area covered by this newspaper boy could be modelled by a Poisson ran
dom variable with mean A = 10. Find the probability mass function of
the random variable that represents the daily proﬁt for a boy who used
to buy 50 papers each day. [10 marks] Total: [22 marks] 4. Suppose the joint probability density function of the continuous random vari—
ables X and Y is given by cxe'zWH) if 0 < a: < +00, 0 < y < +00
ﬂay) = 0, otherwise where c is a positive constant. (a) Show that c = 1. [3 marks] S&AS: STAT1301 Statistics and Probability I 3 (b) Find the marginal probability density functions fX(a:) and fy(y) of X and Y, respectively. [6 marks]
(c) State, with reasons, whether X and Y are independent. [3 marks]
(d) Find E(X), Var(X) and E(X Y). Does E(Y) exist? Give reasons to
support your conclusion. [8 marks] Total: [20 marks] 5. (a) The mean breaking strength of a certain type of cord has been established
from considerable experience at 18.3 ounces with a standard deviation of
1.2 ounces. A new machine is purchased to manufacture this type of cord.
A sample of 100 pieces obtained from the new machine shows a mean
breaking strength of 17.0 ounces. The standard deviation of this type of
cord produced by this new machine is assumed to remain unchanged at
1.2 ounces. Use an appropriate test to see whether the products produced
are acceptable or not, using the 1% level of signiﬁcance. State clearly your
ﬁndings. [6 marks] (b) Two independent random samples, each of size 10, from two independent
normal population random variables X and Y, where X N N(p.1, 02) and
Y ~ N012, 02) yield the sample means and sample variances as i = 4.8,
3% = 8.64, g = 5.6 and 5% = 7.88. Find a 95 percent conﬁdence interval
for [11 — pg. [8 marks] (c) Two surveys were independently conducted to estimate a population
mean, a. Denote the estimators by X1 and X2 and the standard de
viations of the estimators by $1 and 52, respectively. Assume that X1
and X2 are both unbiased. For some a and ﬂ, the two estimators can be
combined to give a better estimator: X =OZX1 +,BX2 (i) Find the conditions on a and ﬂ that make the combined estimator
unbiased. [4 marks] (ii) Find the values of a and ﬂ which minimize the variance of the
combined estimator, subject to the condition of unbiasedness. [4 marks]
Total: [22 marks] ...
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 Spring '08
 SMSLee

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