Arc sine Distribution. Let U = sin2θ, where θ has a uniform distribution on [0, 2π]. Verify that P{U ≤ u} = arcsin √u, u ∈ [0, 1], which is the arc sine distribution. Let X 1 , X 2 be independent...

Arc sine Distribution. Let U = sin2θ, where θ has a uniform distribution on [0, 2π]. Verify that P{U ≤ u} = arcsin √u, u ∈ [0, 1], which is the arc sine distribution. Let X₁, X₂
be independent normally distributed random variables with mean 0 and variance 1. Show that X²
₁
/(X²
₁
+ X²
₂
) d = U.