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2.2.3 Expectation Value, Variance, Functions of r.v.'s

Now that we have assigned a number to the outcome of an event, we can define an ``average'' value for the r.v. over the possible events. This average value is called the expectation value for the random variable x, and has the following definition:

One can define a unique, real-valued function of a r.v., which will also be a r.v. That is, given a r.v. x, then the real-valued function is also a r.v. and we can define the expectation value of :

The expectation value of a linear combination of r.v.'s is simply the linear combination of their respective expectation values;

The expectation value is simply the ``first moment'' of the r.v., meaning that one is finding the average of the r.v. itself, rather than its square or cube or square root. Thus the mean is the average value of the first moment of the r.v., and one might ask whether or not averages of the higher moments have any significance. In fact, the average of the square of the r.v. does lead to an important quantity, the variance, and we will now define the higher moments of a r.v. x as follows: