Arcsine distribution

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Arcsine
Probability density function
Probability density function for the arcsine distribution
Cumulative distribution function
Cumulative distribution function for the arcsine distribution
Parameters none
Support
PDF
CDF
Mean
Median
Mode
Variance
Skewness
Ex. kurtosis
Entropy
MGF
CF

In probability theory, the arcsine distribution is the probability distribution whose cumulative distribution function is

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 the standard arcsine distribution then

The arcsine distribution appears

Generalization[edit]

Arcsine – bounded support
Parameters
Support
PDF
CDF
Mean
Median
Mode
Variance
Skewness
Ex. kurtosis

Arbitrary bounded support[edit]

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 (ab).

Shape factor[edit]

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[edit]

  • Arcsine distribution is closed under translation and scaling by a positive factor
    • If
  • The square of an arc sine distribution over (-1, 1) has arc sine distribution over (0, 1)
    • If

Differential equation

Related distributions[edit]

  • If U and V are i.i.d 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

See also[edit]

References[edit]