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Application Example 11
(Functions of several random variables)
(Note that the Extreme Type Distribution will be covered in more detail in lectures
relating to distribution models)
DISTRIBUTION OF THE MAXIMUM OF INDEPENDENT
IDENTICALLYDISTRIBUTED VARIABLES
Many engineering applications require the calculation of the distribution of the maximum
of a number n of indendent, identically distributed (iid) variables. A typical situation is
the design of a system for the “nyear demand” when the maximum demands in different
years are iid (design of a dam for the nyear flood, design of an offshore platform for the
nyear wave, design of a building for the nyear wind, etc.).
In some cases, for example the design of buildings against earthquake loads,
using the year as the basic unit of time makes little sense, since earthquake occurrences
do not have a yearly cycle (floods, winds, and seastates do). Rather, earthquakes may be
viewed as occurring at random times, say according to a Poisson process and
maximization of the quantity of interest (for example earthquake magnitude or the
induced monetary loss) should be done over the random number of earthquakes in a time
period of duration T. Accordingly, we consider below maximum problems of two types:
Y
1
= max{X
1
, X
2
, .
.., X
n
}
(
1
a
)
Y
2
= max{X
1
, X
2
, .
.., X
N
}
(1b)
where n in Eq. 1a is fixed (e.g. number of years) and N in Eq. 1b is a random variable
with Poisson distribution and mean value
λ
T.
Maximum of a fixed number n of iid variables (Eq. 1a)
1
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View Full DocumentLet F
X
(x) be the common distribution of the variables X
i
in Eq. 1a and let F
n
(y) be the
corresponding distribution of Y
1
= max{X
1
, X
2
, .
.., X
n
}. Obtaining F
n
(y) from F
X
(x) is
very simple. In fact,
F
n
(y) = P[(X
1
≤
y)
∩
(X
2
≤
y)
∩
...
∩
(X
n
≤
y)] = {F
X
(y)}
n
(2)
Therefore, the CDF of Y
1
is obtained by taking the n
th
power of the CDF of the X
i
.
This result suffices when the distribution F
X
is accurately known. In some cases,
F
X
is not completely known. It is then of interest to see whether, for large n, the
distribution of Y
1
approaches a standard shape, which does not depend on F
X
. Theoretical
analysis shows that this indeed happens, but that the distribution F
n
(y) for n large is not
entirely independent of F
X
. One important result is that the distribution of Y
1
approaches
a socalled Extreme Type 1 (EX1) distribution if the probability density of X decays in
the upper tail as an exponential function. This includes exponential, normal, lognormal
and gamma F
X
distributions, among others. A second result is that, if the upper tail of X
decays as a power function of x, then the distribution of Y
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 Fall '07
 DonHill

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