Quiz # 2
Inclass, open books and notes
Problem 1 (40 Points)
An environmental variable X has value X
i
in day i. Due to budgetary constraints, X is
measured only every third day. If X is measured in day i, then the observed value of X
i
1
is used to estimate the threeday average
X
=
3
(X
i
−
1
+
X
i
+
X
i
+
1
) . Given that the
variables X
i
have multivariate normal distribution with common mean value m and
common variance
σ
2
and that the correlation coefficient between X
i
and
X
j
is
ρ
ij
=
0.9
i
−
j
,
(a) Find the distribution of
(X X
i
)
. [Hint: start by finding the joint distribution of
X
and X
i
].
(b) Based on the answer to (a), suggest a better estimator of X from X
i
.
Problem 2 (30 Points)
In a river reach, flooding occurs if the water level H exceeds 8 meters. Following a heavy
rainstorm, the water level is
H
= H
o
+
D
I
5
.
0
8
.
0
, where H
o
is the water level before the
storm, I is storm depth in meters/hour (integrated over the river catchment) and D is the
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 Spring '05
 DanieleVeneziano
 Normal Distribution, Standard Deviation, Probability theory, mean value

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