5.2: Central Role of the Gaussian and Rayleigh Distributions

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Page ID: 47245

Franz S. Hover & Michael S. Triantafyllou
Massachusetts Institute of Technology via MIT OpenCourseWare

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The Central Limit Theorem - which states that the sum of a large number of random variables approaches a Gaussian - ensures that stationary and ergodic processes create a data trace that has its samples normally distributed. For example, if a histogram of the samples from an ocean wave measurement system is plotted, it will indicate a normal distribution. Roughly speaking, in any given cycle, the trace will clearly spend more time near the extremes and less time crossing zero. But for the random process, these peaks are rarely repeated, while the zero is crossed nearly every time. It is recommended to try a numerical experiment to confirm the result of a normal distribution:

\[ p(y) = \dfrac{1}{\sqrt{2 \pi} \sigma_y} e^{-y^2 / 2 \sigma_y^2}, \]

where the standard deviation is \(\sigma_y\) and the mean is zero. As indicated above, the standard deviation is precisely the square root of the area under the one-sided spectrum.

In contrast with the continuous trace above, heights are computed only once for each cycle. Heights are defined to be positive only, so there is a lower limit of zero, but there is no upper limit. Just as the signal \(y\) itself can theoretically reach arbitrarily high values according to the normal distribution, so can heights. It can be shown that the distribution of heights from a Gaussian process is Rayleigh:

\[ p(h) = \dfrac{h}{4 \sigma_y^2} e^{-h^2 / 8 \sigma_y^2}, \]

where \(\sigma\) here is the standard deviation of the underlying normal process. The mean and standard deviation of the height itself are different:

\begin{align} \bar{h} \, &= \sqrt{2 \pi} \, \sigma_y \simeq 2.5 \sigma_y \\[4pt] \sigma_h \, &= \sqrt{8 - 2\pi} \, \sigma_y \simeq 1.3 \sigma_y. \end{align}

Notice that the Rayleigh pdf has an exponential of the argument squared, but that this exponential is also multiplied by the argument; this drives the pdf to zero at the origin. The cumulative distribution is the simpler Rayleigh cpf:

\[ p(h < h_o) \, = \, 1 - e^{- h_o^2 / 8 \sigma_y^2}; \]

\(P(h)\) looks like half of a Gaussian pdf, turned upside down! A very useful formula that derives from this simple form is that

\[ p(h > h_o) \, = \, 1 - p(h < h_o) \, = \, e^{-2 h_o^2 / (\bar{h}^{1/3})^2} . \]

This follows immediately from the cumulative probability function, since \( \bar{h}^{1/3} = 4 \sigma_y \). It is confirmed that \( p(h > \bar{h}^{1/3}) = e^{-2} \simeq 0.13. \)