6-ConceptsProbability Page 187 Wednesday, February 4, 2004 3:00 PM
Concepts of Probability 187EX[ k]= ∑x pki i
respectively. If the variable Xis continuous and has a density p(x) such
that∞∫ px()dx=^1
- ∞
we can write∞∫
p
EX[
p
]= x px()dx- ∞
and∞EX[ k]= ∫x pxk ()dx
- ∞
respectively.
The centered moments are the moments of the fluctuations of the
variables around its mean. For example, the variance of a variable Xis
defined as the centered moment of second order:()= σ
2
var X = σ()= EX X ) ]
2
X [( –
2
x
∞ ∞ ∞ 2
(xX)
2- px()dx= x ()dx–
2
= ∫ ∫ px ∫ xp x()dx
- ∞ –∞ –∞
where X= EX[].
The positive square root of the variance, σxis called the standard
deviation of the variable.
We can now define the covariance and the correlation coefficient of
a variable. Correlation is a quantitative measure of the strength of the
dependence between two variables. Intuitively, two variables are depen-
dent if they move together. If they move together, they will be above or
below their respective means in the same state. Therefore, in this case,
the product of their respective deviations from the means will have a
positive mean. We call this mean the covariance of the two variables.