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EC2019
JANUARY EXAMINATIONS 2009
Subject
ECONOMICS
Title of Paper
EC2019 SAMPLING AND INFERENCE
Time Allowed
TW0 HOURS
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Instructions to candidates
Answer THREE questions
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Page 1 of 4
CONTINUED …
EC2019
1. By invoking the rules of Boolean algebra, prove that, if A and B are two
events, then the probability that at least one of them will occur is given
by
P (A ∪ B) = P (A) + P (B) − P (A ∩ B).
The following diagram represents three components in an electrical circuit
which are protected by fuses:
A
C
B
For the current to flow, the fuse C must remain intact—the event C—
and fuse A or fuse B or both must remain intact—the event A ∪ B. The
probabilities that the fuses will blow are independent and are given by
P (Ac ) = 2/3, P (B c ) = 1/3, P (C c ) = 1/4. Calculate the probability the
the current will flow.
2. Let H denote an hypothesis and let E denote an event. Show that
P (Hi |E) =
where P (E) =
i
P (E|Hi )P (Hi )
P (E)
P (E|Hi )P (Hi ).
Of the tins in a paint store, 60% contain Brilliant White paint and 40%
contain Off-White paint. The tins are unmarked. However, 23 of the Brilliant White paint and 13 of the Off-White paint is delivered in tins without
handles.
(a) What is the probability that a tin will contain Brilliant White paint
given that it has a handle?
(b) What is the probability that, by picking a tin with a handle and another without a handle, the storekeeper will select one of each colour?
CONTINUED
Page 2 of 4
EC2019
3. Demonstrate how the moments of a random variable x may be obtained
from its moment generating function by showing that the rth derivative of
E(ext ) with respect to t gives the value of E(xr ) at the point where t = 0.
Demonstrate that the moment generating function of a sum of independent
variables is the product of their individual moment generating functions.
Find the moment generating function of the point binomial
f (x; p) = px (1 − p)1−x
where x = 0, 1. What is the relationship between this and the m.g.f. of
the binomial distribution?
Find the variance of x1 +x2 when x1 ∼ f (p1 = 0.25) and x2 ∼ f (p2 = 0.75)
are independent point binomials.
4. Derive the binomial distribution by considering the sum of the outcomes
xi ; i = 1, . . . , n of n independent trials where P (xi = 1) = p and P (xi =
0) = 1 − p for all i.
Rocket A has four motors and rocket B two motors. The probability of
the failure of an individual motor is p. Rocket A will fail if three or more
motors fail, and Rocket B will fail if both of its motors fail.
(i) Find the probability that rocket A will fail.
(ii) Find the probability that rocket B will fail.
(iii) Find the probability p of the failure of an individal motor on the
assumption that rocket A and rocket B have the same probability of
failure.
CONTINUED
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EC2019
5. Let x and y be jointly distributed random variables such that E(y|x) =
α + βx. Prove that β = C(x, y)/V (x) and that α = E(y) − βE(x)
The average height of each generation of adult males in Britain is 69 ins
with a standard deviation of 5 ins. The correlation coefficient for the
heights of fathers and the heights of their sons is 0.6. Given that I am 74
ins tall, what is the expected height of my male offspring?
6. Describe (a) the difference between a one-tailed test of a statistical hypothesis and a two-tailed test, and (b) the difference between a Type I
error and a Type II error.
A manufacturer claims to be filling his glass jars with 200 grammes of
instant coffee. The weights and measures department of the Office of
Fair Trading have taken a sample of 49 jars (they meant to take 50 but
they broke one jar) and have found an average weight of 195 grammes
and a standard deviation of 5 grammes. Is there any evidence that the
manufacturer is systematically underfilling his jars?
END OF PAPER
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