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ORIE 3500/5500 Fall Term 2008
Assignment 10
Note that
•
the due date is
Monday, November 24
, at 1:00 pm
•
you need to mention your section number on the top of your answer
script.
1. [5 + 5 + 5 = 15 pts]
The points (
X, Y
) obtained by the students of a course in the two
prelims is modeled by a bivariate normal distribution with parameters
μ
X
= 60
, μ
Y
= 75
, σ
X
= 18
, σ
Y
= 12
, ρ
= 0
.
75. Maximum possible
points in both the prelims are 100.
(a) If the ﬁnal score consists of 40% of prelim 1 and 60% of prelim 2,
then ﬁnd the distribution of the ﬁnal score.
(b) If there are 300 students enrolled for the course, how many of
them are expected to score more than 70 in prelim 1?
(c) If 50 students score 70 in prelim 1, how many of them are expected
to score more than 80 in prelim 2?
2. [5 + 5 = 10 pts]
Suppose
X
∼
Poisson
(
λ
) and
α
∈
(0
,
1).
(a) Apply the Markov inequality for the random variable
Y
s
=
e

sX
to get an upper bound for
P
(
X < αλ
).
(b) Minimize the bound obtained in (a) with respect to
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 Fall '08
 WEBER

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