9.9 Extreme-Value Theory
133
to exceed a certain large value? Furthermore, we might be curious about the distribution of extreme values. This is basically the histogram of values that exceed all
previous samples. We will discuss both topics but start with the first question and
follow [9] in doing so.
So, first we ask ourselves how fast the value of the maximum sample grows with
the number of drawings n. We start from a normalized distribution function D(x)
and its cumulative distribution function
C(x) =
x
ˆ
x
D(x
)dx
,
(9.31)
where ˆ
x is the minimum possible value, which could be −∞, for example, for a
Gaussian distribution. For a power law distribution it would be the minimum value,
for example 1. Note that we could equally well have introduced the tail fraction
T (x) = 1 − C(x) which is the area under the distribution function from x to the
maximum value, which often is infinity. The probability to find a sample between x
and x + dx after n drawings is given by
(x)dx = nC(x)
n−1 D(x)dx
(9.32)
because n − 1 times we have to find a value smaller than x, which accounts for the
factor C(x)
n−1 and once a value between x and x + dx, which accounts for D(x)dx.
Moreover, there are n different ways to choose when the maximum is drawn.
The sought-after expected maximum value after n drawings can therefore be
calculated as the expectation value of x with respect to the distribution function
(x)
x n =
∞
ˆ
x
x(x)dx = n
∞
ˆ
x
xC(x)
n−1 D(x)dx .
(9.33)
Let us now calculate x n for different distributions D(x), in order to explore their
scaling for large n. First, we consider an exponential distribution, defined on the
interval from 0 to infinity
D(x)dx = e
−x dx such that C(x) = 1 − e
−x
.
(9.34)
The expectation value for the maximum after n drawings is then
x n exp = n
∞
0
xe
−x
(1 − e
−x
)
n−1 dx ,
(9.35)
which we integrate numerically for n = 10
m with m = 1, . . . , 10 and show the result
as the solid black line on Fig. 9.10. We observe that x n exp grows linearly with log(n).
133
to exceed a certain large value? Furthermore, we might be curious about the distribution of extreme values. This is basically the histogram of values that exceed all
previous samples. We will discuss both topics but start with the first question and
follow [9] in doing so.
So, first we ask ourselves how fast the value of the maximum sample grows with
the number of drawings n. We start from a normalized distribution function D(x)
and its cumulative distribution function
C(x) =
x
ˆ
x
D(x
)dx
,
(9.31)
where ˆ
x is the minimum possible value, which could be −∞, for example, for a
Gaussian distribution. For a power law distribution it would be the minimum value,
for example 1. Note that we could equally well have introduced the tail fraction
T (x) = 1 − C(x) which is the area under the distribution function from x to the
maximum value, which often is infinity. The probability to find a sample between x
and x + dx after n drawings is given by
(x)dx = nC(x)
n−1 D(x)dx
(9.32)
because n − 1 times we have to find a value smaller than x, which accounts for the
factor C(x)
n−1 and once a value between x and x + dx, which accounts for D(x)dx.
Moreover, there are n different ways to choose when the maximum is drawn.
The sought-after expected maximum value after n drawings can therefore be
calculated as the expectation value of x with respect to the distribution function
(x)
x n =
∞
ˆ
x
x(x)dx = n
∞
ˆ
x
xC(x)
n−1 D(x)dx .
(9.33)
Let us now calculate x n for different distributions D(x), in order to explore their
scaling for large n. First, we consider an exponential distribution, defined on the
interval from 0 to infinity
D(x)dx = e
−x dx such that C(x) = 1 − e
−x
.
(9.34)
The expectation value for the maximum after n drawings is then
x n exp = n
∞
0
xe
−x
(1 − e
−x
)
n−1 dx ,
(9.35)
which we integrate numerically for n = 10
m with m = 1, . . . , 10 and show the result
as the solid black line on Fig. 9.10. We observe that x n exp grows linearly with log(n).