Following Chapter 3, we define the *Q*-function as

which represents *the area under the tail of the standard Gaussian distribution*. In this appendix, we derive some useful bounds on the *Q*-function for large positive values of *x*.

To this end, we change the variable of integration in (B.1) by setting

and then recast (B.1) in the form

For any real *z*, the value of exp (−1/2*z*^{2}) lies between the successive partial sums of the power series:

Therefore, for *x* > 0 we find that, on using (*n* + 1) terms of this series, the *Q*-function lies between the values taken by the integral

for even *n* and odd *n*. We now make another change in the integration variable by setting

and also use the definite integral

Doing so, we obtain the following asymptotic expansion for the ...

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