Mean and Variance of a Probability Law

It has been emphasized that in order to describe a numerical valued random phenomenon one must specify its probability function \(P[\cdot]\) or, equivalently, its distribution function \(F(\cdot)\) . In the special case in which the random phenomenon obeys a discrete or a continuous probability law its probability function is determined by a knowledge of the probability mass function \(p(\cdot)\) or of the probability density function \(f(\cdot)\) . Thus, to describe a numerical valued random phenomenon, certain functions must be specified. It is desirable to be able to summarize some of the outstanding features of the probability law of a numerical valued random phenomenon by specifying only a few numbers rather than an entire function. Such numbers are provided by the expectation of various functions \(g(\cdot)\) with respect to the probability law of the random phenomenon.