A2.1 Log-normal Distribution

Further to Chapter 11, Section 11.7.3, probability is

where

• ? = standard deviation

• x _m = median x (half events are < median and half > median)

• mean/median = exp .

Using logarithmic abscissa

The mean, ?, and median, ? _m, are related to the standard deviation, ? ₀, by

A2.2 Rayleigh Distribution

Further to Chapter 12, Section 12.3.3, the mean [1] = = 1.2533 ? and lies at 54.4 per cent of the cumulative distribution; median = = 1.1774 ?. The mode is the most frequently occurring observation and equals ?.

Shearman ^[1] points out that if we put p( v) in terms of the instantaneous power, w = v ², then p( w) dw must equal p( v) d v, whence

Hence,

where w ₀ is the mean value of the power, w, which in the notional 1 ? is the same as the variance of v, which is ? ². The above equation is the exponential probability density distribution of w. Hence, this is sometimes called the Rayleigh-power probability density function (PDF) [2]. However, the Rayleigh PDF is not the same as the exponential PDF, which is:

Rayleigh distribution is the probability of each of the two parts of a bivariate Gaussian distribution and is a...