Civil Engineering Reference
In-Depth Information
Equivalent definitions apply to a discrete random variable
X
. It is in the following
assumed that each realisation
X
of
X
has the same probability of occurrence, and
thus, the mean value and variance of
X
may be estimated from a large data set of
N
individual realisations:
N
1
⎫
∑
x
lim
X
=
⎪
⎪
⎬
⎪
k
N
N
→∞
k
1
=
(2.5)
N
1
2
()
2
∑
(
)
Var X
lim
X
x
==
σ
−
x
k
⎪
⎭
N
N
→∞
k
=
1
The square root of the variance,
σ
, is called the standard deviation. Recalling that
x
[]
E Xx
=
, the expression for the variance may be further developed into
⎡
2
⎤
⎡
2
(
)
2
2
2
2
⎤ ⎡ ⎤
σ
=
E
Xx
−
=
EX
−
2
xXx
+
=
EX
−
x
(2.6)
x
⎢
⎥ ⎣
⎦ ⎣ ⎦
⎣
⎦
There are three probability density distributions that are of primary importance in wind
engineering. These are the Gaussian (normal), Weibull and Rayleigh distributions, each
defined by the following expressions:
⎫
⎡
2
⎤
1
1
⎛
xx
⎞
−
()
⎪
px
exp
⎢
⎥
=
−
⎜
⎟
2
2
⎢
σ
⎥
⎪
πσ
⎝
⎠
x
x
⎣
⎦
⎪
⎪
⎡
β
⎤
β
−
1
x
x
⎛⎞
⎪
()
px
=
β
exp
⎢
−
⎥
(2.7)
⎬
⎜⎟
β
γ
γ
⎢
⎥
⎝⎠
⎪
⎣
⎦
⎪
2
⎡
⎤
⎪
x
1
⎛⎞
x
()
px
exp
⎢
⎥
=
−
⎪
⎜⎟
2
2
γ
γ
⎢
⎝⎠
⎥
⎪
⎣
⎦
⎭
They are graphically illustrated in Fig. 2.1. It is seen that a Rayleigh distribution is the
Weibull distribution with
β
=2.
