...expectations.

For example, the simplest such definition is that your prevision for a random quantity X is the value x that you consider to be a ``fair price'' for a ticket which pays X.

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... .

Strictly, the inner product space is defined over the closure of the equivalence classes of random quantities which differ by a constant, so that we identify any vector, such as , with zero variance with the zero vector.

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...rank

If 29#24 is not of full rank, then we may discard elements of D so that the reduced collection is of full rank. Otherwise, we may consider 30#25 to be the Moore-Penrose generalised inverse in the following matrix equations.

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...beliefs.

As an example of the type of simple rule of thumb that might sometimes be of use, observe that were all the elements of D to be normally distributed, then it would follow that

134#116

In certain circumstances, we might find it useful to approximate the distribution of 135#117, for example by a distribution of form cX, where c is a constant and X has a distribution, with degrees of freedom. Matching the mean and variance suggests a choice of 138#118 degrees of freedom and 139#119.

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...[B]

for example, will automatically be bounded if D has a finite number of elements

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