of one variable is large compared to its mean value the outcome of the other variable is likely
to be small compared to its mean value. A positive correlation coefficient between two
variables implies that if the outcome of one variable is large compared to its mean value the
outcome of the other variable is also likely to be large compared to its mean value. If two
variables are independent their correlation coefficient is zero and the joint density function is
the product of the 1-dimensional density functions. In many cases it is possible to obtain a
sufficiently accurate approximation to the n-dimensional cumulative distribution function
from the 1-dimensional distribution functions of the n variables and their parameters, and the
correlation coefficients.
If Y is a linear function of the random vector X(XX 12 , ,...,Xn)T i.e.:
0
1n
ii
iYa aX
(2.44)
using Equation (2.37), (2.38) and Equation (2.42) it can be shown that the expected value
EY and the variance Var Y are given by:
0
12
1,1(^2) ij
n
ii
i
nn
ii ijXX
iijij
EY a aE X
Var Y a Var X a a C
/
"
"#"
$%
! (2.45)
Conditional Distributions and Conditional Moments
The conditional probability density function for the random variableX 1 , conditional on the
outcome of the random variable X 2 is denoted fXX 12 (xx 12 ) and defined by:
12
12
2,12
12
2(, )
()
()
XX
XX
Xf xx
fxx
fx(2.46)
in accordance with the definition of conditional probability given previously.
As for the case when probabilities of events were considered two random variables X 1 and
X 2 are said to be independent when:
fXX 12 () (xx 12 f xX 1 1 ) (2.47)
By integration of Equation (2.46) the conditional cumulative distribution FxxXX 12 () 12 is
obtained:
1
12
12
2,2
12
2(, )
()
()
x
XX
XX
Xf zx dz
Fxx
fx
(2.48)and finally by integration of (2.48) weighed with the probability density function of X 2 , i.e.
fX 1 ()x 1 the unconditional cumulative distribution is achieved by the total probability
theorem:
FxX 1 () 1