
Probability and Statistics Overview 23
related to the first and second moments through the relationship
Var(X) = E
(X −E(X))
2
= E
X
2
− 2XE(X) + (E(X))
2
= E(X
2
) − (E(X))
2
.
(2.46)
One of the useful concepts in understanding the variation of a random
variable X is the coefficient of variation (CV), which is defined as the r atio of
the standard deviation to the mean (that is, CV =
p
Var(X)/E(X)). It is the
inverse of the so-calle d signal-to-noise ratio. A random variable with CV < 1
is considered to have low variatio n, while one with CV > 1 is considered to
have high variation.
Note that the moments of a function of a random variable, Y = η(X), can
be defined