Moments of random variables: a system-theoretic interpretation

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