Civil Engineering Reference
In-Depth Information
Chapter 2
SOME BASIC STATISTICAL CONCEPTS
IN WIND ENGINEERING
2.1 Parent probability distributions, mean value and variance
()
p x
is defined by
For a continuous random variable
X
, its probability density function
()
dP x
[
]
(
)
( )
()
Pr
x
X x dx
Px dx
Px
dx pxdx
≤≤+ =
+ −
=
=
(2.1)
dx
()
P x
is the cumulative probability function, from which it follows that
where
x
[
]
()
()
Pr
X
xPx
∫
px x
≤=
=
(2.2)
−∞
()
lim
Px
1
and that
=
. Similarly, for two random variables
X
and
Y
the joint probability
x
→∞
density function is defined by
(
)
2
dP xy
,
(
)
pxy
,
=
(2.3)
dx dy
(
)
[
]
where
P xy
,
=
r
X
≤
xY
,
≤
y
. The mean value and variance of
X
are given by
+∞
⎫
[]
()
∫
xEX
=
=
xpx x
⋅
⎪
⎪
⎬
⎪
−∞
(2.4)
+∞
2
2
()
2
⎡
(
)
⎤
(
)
( )
Var X
E
X
x
∫
x
x
p x dx
==
σ
−
=
− ⋅
⎪
⎢
⎥
x
⎣
⎦
⎭
−∞
