Arcsine distribution
Probability density function | |||
Cumulative distribution function | |||
Parameters | none | ||
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Support | |||
CDF | |||
Mean | |||
Median | |||
Mode | |||
Variance | |||
Skewness | |||
Excess kurtosis | |||
Entropy | |||
MGF | |||
CF |
In
for 0 ≤ x ≤ 1, and whose probability density function is
on (0, 1). The standard arcsine distribution is a special case of the beta distribution with α = β = 1/2. That is, if is an arcsine-distributed random variable, then . By extension, the arcsine distribution is a special case of the Pearson type I distribution.
The arcsine distribution appears in the
Generalization
Parameters | |||
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Support | |||
CDF | |||
Mean | |||
Median | |||
Mode | |||
Variance | |||
Skewness | |||
Excess kurtosis | |||
CF |
Arbitrary bounded support
The distribution can be expanded to include any bounded support from a ≤ x ≤ b by a simple transformation
for a ≤ x ≤ b, and whose probability density function is
on (a, b).
Shape factor
The generalized standard arcsine distribution on (0,1) with probability density function
is also a special case of the beta distribution with parameters .
Note that when the general arcsine distribution reduces to the standard distribution listed above.
Properties
- Arcsine distribution is closed under translation and scaling by a positive factor
- If
- The square of an arcsine distribution over (-1, 1) has arcsine distribution over (0, 1)
- If
- The coordinates of points uniformly selected on a circle of radius centered at the origin (0, 0), have an distribution
- For example, if we select a point uniformly on the circumference, , we have that the point's x coordinate distribution is , and its y coordinate distribution is
Characteristic function
The characteristic function of the generalized arcsine distribution is a zero order Bessel function of the first kind, multiplied by a complex exponential, given by . For the special case of , the characteristic function takes the form of .
Related distributions
- If U and V are uniform(−π,π) random variables, then , , , and all have an distribution.
- If is the generalized arcsine distribution with shape parameter supported on the finite interval [a,b] then
- If X ~ Cauchy(0, 1) then has a standard arcsine distribution
References
- ISBN 978-1-5386-0595-0.
- .
Further reading
- Rogozin, B.A. (2001) [1994], "Arcsine distribution", Encyclopedia of Mathematics, EMS Press