Probability, Random Variables, and Random Processes: Theory and Signal Processing Applications

E.9 COMPLETING THE SQUARE

Consider the quadratic equation

(E.57)

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 ...