Page 348 - Numerical Methods for Chemical Engineering

Random vectors and multivariate distributions 337

A related concept is the correlation of X and Y, corr (X, Y) = cov(X, Y)/ var(X) + var(Y).
If X and Y are independent, cov(X, Y) = 0; however, a covariance of zero does not neces-
sarily imply that the two variables must be independent (although it suggests that they are).
If cov(X, Y) > 0, then when X is greater than its mean E(X), Y tends also to be greater than
its mean E(Y). Conversely, if cov(X, Y) < 0, then if X > E(X), it is more probable that Y
is less than E(Y). A nonzero covariance means only that the two variables tend to behave in
a related manner, it does not mean that there is a cause and effect relationship among them.
Asserting the latter is a common fallacy.
Definition Covariance matrix of a random vector
Let v be a vector whose components are random variables, not necessarily independent.
Then, the covariance matrix of v,cov(v), has elements

[cov(v)] ij = E{[v i − E(v i )][v j − E(v j )]} (7.108)
If each component of v is independent of all others, cov(v) is diagonal,
 
var(v 1 )
 var(v 2 ) 
 
. (7.109)
cov(v) =  . 
 . 
var(v N )
2
If in addition, each component of v has the same variance σ , then
2
cov(v) = σ I (7.110)
If v = Ax, where A is a constant matrix and x is another random vector, then
cov(v) = cov(Ax) = A[cov(x)]A T (7.111)

If v is a random vector and c is a constant vector,
T T
var(c · v) = var c v = c [cov(v)]c = c · [cov(v)]c (7.112)
The covariance matrix is always symmetric and positive-definite.
Definition Multivariate Gaussian (normal) distribution
Let v be a random N-dimensional vector with a mean µ = E(v) and a covariance matrix
. Since is symmetric, positive-definite, −1 always exists. The Gaussian (normal)
distribution of v is
1
1 1
−1
T
P(v; µ, ) = √ exp − (v − µ) (v − µ) (7.113)
(2π) N/2 | | 2
The Boltzmann and Maxwell distributions

Many applications of probability theory to chemical engineering arise in statistical mechan-
ics, the microscopic theory that underpins thermodynamics. Consider a system whose state
is described by the state vector q, such that the energy in this microstate is E(q). A key result
of statistical mechanics is the Boltzmann distribution. For a system closed to its surround-
ings with respect to the exchange of mass, held at a constant temperature T and volume V,