TABLE OF CONTENTS The Gaussian probability density function is used to describe the statistical behavior of a random variable or noise source with zero mean and mean-squared value of ? 2. It describes the probability of the signal having any specified instantaneous value.
Of more relevance to us is the probability of a noise signal exceeding a threshold value. This function is defined by erfc (x); see Table 8.3. Since the Gaussian distribution is symmetrical and the total probability of a signal across its full amplitude range is 1, then the probability of a noise signal exceeding its mean is erfc(O)=0.5.
Erfc (x) can be used to find the probability of a noise signal exceeding any value, remembering that ?, the standard deviation of the Gaussian variable, is the rms value of the noise signal. For example, what is the probability of a noise signal exceeding the peak level of a sine wave of the same power? Since the powers are the same, V rms (sinewave)= V rms (noise). Also, V peak (sine) = 1.414 V rms. Therefore, look up erfc(1.414) to find the probability of 7.8%.
If we have N independent random variables, each with the same mean and variance, then the composite signal equal to the sum of these variables is the chi-squared variable with N degrees of freedom. The probability density function (pdf) of chi-squared is tabulated in Table 8.5.
