📂Probability Distribution

Rayleigh Distribution

Definition ¹

A continuous probability distribution with the probability density function given for the scale parameter $\sigma > 0$ is called the Rayleigh Distribution. $$ f(x) = {{ x } \over { \sigma^{2} }} e^{ - x^{2} / (2 \sigma^{2})} \qquad , x \ge 0 $$

Theorem

• [1]: If $X, Y \sim N \left( 0, \sigma^{2} \right)$ then $\sqrt{X^{2} + Y^{2}}$ follows a Rayleigh Distribution with $\sigma > 0$.

Explanation

As can be seen from Theorem [1], the Rayleigh Distribution is the distribution followed by the magnitude of a probability vector following a bivariate normal distribution. It is also known as a special case of the Weibull distribution.