10.1 RAYLEIGH DISTRIBUTION CHARACTERISTICS

The Rayleigh pdf is given by [1, p. 100]

(10.1)

? is the scale parameter and the only parameter in this distribution. Figure 10.1 shows how the shape of the Rayleigh pdf changes as ? changes from ? = 0.5 to ? = 2.0.

Figure 10.1: Rayleigh pdf's for various values of the scale parameter, ?.

Some of the specific characteristics of the Rayleigh pdf are the following:

1. The Rayleigh pdf is a special case of the Weibull pdf with ? = 2 and , where ?' is the scale parameter of the Weibull pdf and ? is the scale parameter of the Rayleigh pdf in Eq. (10.1).

2. As the value of ? increases, the Rayleigh pdf becomes stretched out to the right.

3. The mean, T, of the Rayleigh pdf is given by

   (10.2)

4. The median, ?, of the Rayleigh pdf is given by

   (10.3)

5. The mode, , of the Rayleigh pdf is given by

   (10.4)

6. The standard deviation, ? _T, of the Rayleigh pdf is given by

   (10.5)

7. The coefficient of skewness, ? ₃, of the Rayleigh pdf is

   (10.6)

8. The coefficient of kurtosis, ? ₄, of the Rayleigh pdf is

   (10.7)

9. The kth moment of the Rayleigh pdf about the origin is given by

   (10.8)

10. The kth moment of the Rayleigh pdf