Computation of probabilities that involve a Gaussian process require finding the area under the tail of the Gaussian (normal) probability density function as shown in Figure F.1.

Figure F.1. Gaussian probability density function. Shaded area is Pr(x ≥ x₀) for a Gaussian random variable.

Figure F.1 illustrates the probability that a Gaussian random variable x exceeds x₀, Pr(x ≥ x₀), which is evaluated as

Equation F.1. 

The Gaussian probability density function in Equation (F.1) cannot be integrated in closed form.

Any Gaussian probability density function may be rewritten through use of the substitution

Equation F.2. 

to yield

Equation F.3. 

where the kernel of the integral on the right-hand side of Equation (F.3) is the normalized Gaussian probability density function with mean of 0 and standard deviation of 1. Evaluation of the integral in Equation (F.3) is designated as the Q-function, which is defined as

Equation F.4. 

Hence Equations (F.1) or (F.3) can be evaluated as

Equation F.5. 

The Q-function is bounded by two analytical expressions as follows:

For values of z greater 3.0, both of these bounds closely approximate Q(z).

Two important properties of Q(z) are

Equation F.6. 

Equation F.7. 

A graph of Q(z) versus z is given in Figure F.2.

Figure F.2. Plot of the Q-function.

A tabulation of the Q-function for various values of z is given in Table F.1.

The error function (erf) is defined as

Equation F.8. 

and the complementary error function (erfc) is defined as

Equation F.9. 

The erfc function is related to the erf function by

Equation F.10. 

The Q-function is related to the erf and erfc functions by

Equation F.11. 

Equation F.12. 

Equation F.13. 

The relationships in Equations (F.11)–(F.13) are widely used in error probability computations. Table F.2 displays values for the erf function.