A continuous
probability distribution
with density
depending on a
scale parameter
.
A Rayleigh distribution has positive asymmetry; its unique mode is at the point
.
All moments of a Rayleigh distribution are
finite, the mathematical expectation and variance being
and
,
respectively. The distribution function of a Rayleigh distribution has the form
A Rayleigh distribution is a special case of the distribution with density
when
;
hence, when
the Rayleigh distribution coincides with the distribution of the
square root of a random variable which has the
"chi-squared" distribution
with two degrees of freedom. In other words, a
Rayleigh distribution can be interpreted as the distribution of the
length of a vector in a plane Cartesian coordinate system,
the coordinates of which are independent and have the
normal distribution
with parameters 0 and
.
In the three-dimensional space the
Maxwell distribution
plays a role analogous to the Rayleigh distribution.
A Rayleigh distribution is mainly applied in target
theory and statistical communication theory. It was first considered by
Lord Rayleigh
in
1880
as the distribution of the
amplitude resulting from the addition of harmonic oscillations.