Moments of continuous random variables with a
probability density function which can be represented as the
impulse response of a linear time-invariant system are studied.
Under some assumptions, the moments of the random variable
are characterised in terms of the solution of a Sylvester equation
and of the steady-state output response of an interconnected
system. This allows to interpret well-known notions and results
of probability theory and statistics in the language of system
theory, including the notion of moment generating function, the
sum of independent random variables and the notion of mixture
distribution.