1.5 Means and variances

1.5.1 Expectations

Suppose that m is a discrete random variable and that the series

is absolutely convergent, that is such that

Then the sum of the original series is called the mean or expectation of the random variable, and we denote it

A motivation for this definition is as follows. In a large number N of trials, we would expect the value m to occur about p(m)N times, so that the sum total of the values that would occur in these N trials (counted according to their multiplicity) would be about

so that the average value should be about

Thus, we can think of expectation as being, at least in some circumstances, a form of very long term average. On the other hand, there are circumstances in which it is difficult to believe in the possibility of arbitrarily large numbers of trials, so this interpretation is not always available. It can also be thought of as giving the position of the ‘centre of gravity’ of the distribution imagined as a distribution of mass spread ...

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