Arcsine distribution

In <a href="/facts/Probability_theory/mFBn51rE">probability theory</a>, the arcsine distribution is the <a href="/facts/Probability_distribution/EpsKKVRu">probability distribution</a> whose <a href="/facts/Cumulative_distribution_function/WaKU8tp4">cumulative distribution function</a> involves the <a href="/facts/Arcsine/bVwU9FPG">arcsine</a> and the <a href="/facts/Square_root/AfrzfBdQ">square root</a>:

F
        (
        x
        )
        =
        
          
            2
            π
          
        
        arcsin
        ⁡
        
          (
          
            
              x
            
          
          )
        
        =
        
          
            
              arcsin
              ⁡
              (
              2
              x
              −
              1
              )
            
            π
          
        
        +
        
          
            1
            2
          
        
      
    
    {\displaystyle F(x)={\frac {2}{\pi }}\arcsin \left({\sqrt {x}}\right)={\frac {\arcsin(2x-1)}{\pi }}+{\frac {1}{2}}}

for 0 ≤ x ≤ 1, and whose <a href="/facts/Probability_density_function/zvfybna4">probability density function</a> is

f
        (
        x
        )
        =
        
          
            1
            
              π
              
                
                  x
                  (
                  1
                  −
                  x
                  )
                
              
            
          
        
      
    
    {\displaystyle f(x)={\frac {1}{\pi {\sqrt {x(1-x)}}}}}

on (0, 1). The standard arcsine distribution is a special case of the <a href="/facts/Beta_distribution/DbJk4eeV">beta distribution</a> with α = β = 1/2. That is, if 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is an arcsine-distributed random variable, then 
 
 
 
 X
 ∼
 
 
 B
 e
 t
 a
 
 
 
 
 (
 
 
 
 
 
 1
 2
 
 
 
 ,
 
 
 
 1
 2
 
 
 
 
 
 )
 
 
 
 
 {\displaystyle X\sim {\rm {Beta}}{\bigl (}{\tfrac {1}{2}},{\tfrac {1}{2}}{\bigr )}}
 
. By extension, the arcsine distribution is a special case of the <a href="/facts/Pearson_distribution/L0oULaWK">Pearson type I distribution</a>.
The arcsine distribution appears in the <a href="/facts/L%25C3%25A9vy_arcsine_law/0VtUvRQ1">Lévy arcsine law</a>, in the <a href="/facts/Erd%25C5%2591s_arcsine_law/0sf42uX1">Erdős arcsine law</a>, and as the <a href="/facts/Jeffreys_prior/6j4Wpu7t">Jeffreys prior</a> for the probability of success of a <a href="/facts/Bernoulli_trial/7fm8snCf">Bernoulli trial</a>. The arcsine probability density is a distribution that appears in several random-walk fundamental theorems. In a fair coin toss <a href="/facts/Random_walk/08v0jmfv">random walk</a>, 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 2N, 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 2N (following the arcsine distribution).

Arcsine distribution open-in-new

Arcsine distribution