app11_max

app11_max - Application Example 11 (Functions of several...

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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 IDENTICALLY-DISTRIBUTED 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 “n-year demand” when the maximum demands in different years are iid (design of a dam for the n-year flood, design of an offshore platform for the n-year wave, design of a building for the n-year 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 sea-states 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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Let 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 so-called 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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app11_max - Application Example 11 (Functions of several...

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