E.9 COMPLETING THE SQUARE
Consider the quadratic equation
which we want to write in the form
(E.58)
It is easily shown that
(E.59)
which is verified by factoring a in (E.57), adding and subtracting b2/4a2, and rearranging the expression as follows:
(E.60)
which has the desired squared term:
(E.61)
This technique is frequently used to rearrange the exponent of a Gaussian pdf.
Example E.2. When integrating a function over the entire range of variable x, often it is convenient to rewrite the integrand as the product of two parts: one in terms of x which has the form of a known pdf, and the other which is independent of x. For example, it may be possible to write
(E.62)
where fX(x) is a valid pdf, and c is a constant or a function of some variable other than x. The parameters of fX(x) may also be a function of another variable, as illustrated next. Suppose we want to integrate ...
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