Type of probability distribution
 ;/math ; |
 cdf =formula_1 |
 mean =formula_2 |
 median =formula_2 |
 mode =formula_4 |
 variance =formula_5 |
 skewness =formula_6|
 kurtosis =formula_7|
 entropy =formula_8 |
 mgf =formula_9|
 char =formula_10|
In probability theory, the arcsine distribution is the probability distribution whose cumulative distribution function involves the arcsine and the square root:
formula_11
for 0 ≤ "x" ≤ 1, and whose probability density function is
formula_12
on (0, 1). The standard arcsine distribution is a special case of the beta distribution with "α" = "β" = 1/2. That is, if formula_13 is an arcsine-distributed random variable, then formula_14. By extension, the arcsine distribution is a special case of the Pearson type I distribution.
The arcsine distribution appears in the Lévy arcsine law, in the Erdős arcsine law, and as the Jeffreys prior for the probability of success of a Bernoulli trial. The arcsine probability density is a distribution that appears in several random-walk fundamental theorems. In a fair coin toss random walk, the probability for the time of the last visit to the origin is distributed as an (U-shaped) arcsine distribution. In a two-player fair-coin-toss game, a player is said to be in the lead if the random walk (that started at the origin) is above the origin. The most probable number of times that a given player will be in the lead, in a game of length 2"N", is not "N". On the contrary, "N" is the least likely number of times that the player will be in the lead. The most likely number of times in the lead is 0 or 2"N" (following the arcsine distribution).
Generalization.
 ;/math ; |
 cdf =formula_15 |
 mean =formula_16 |
 median =formula_16 |
 mode =formula_18 |
 variance =formula_19 |
 skewness =formula_6|
 kurtosis =formula_7|
 entropy = |
 mgf = |
 char = formula_22|
Arbitrary bounded support.
The distribution can be expanded to include any bounded support from "a" ≤ "x" ≤ "b" by a simple transformation
formula_15
for "a" ≤ "x" ≤ "b", and whose probability density function is
formula_24
on ("a", "b").
Shape factor.
The generalized standard arcsine distribution on (0,1) with probability density function
formula_25
is also a special case of the beta distribution with parameters formula_26.
Note that when formula_27 the general arcsine distribution reduces to the standard distribution listed above.
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 formula_22. For the special case of formula_36, the characteristic function takes the form of formula_37.
References.
 ;templatestyles src="Reflist/styles.css" / ;