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.