Question

点随机地落在以原点为圆心,R为半径的圆周上,并对极角是均匀分布的,试求该点的横坐标X的概率密度？

Answer 1

解：极角a服从均匀分布U[0, 2*pi]
设x的分布函数为F(x) = P(X <= x), -R<= x <= R
则F(x) = P(X <= x) = P(cos(a) <= x/R) = P(arccos(x/R)<=a <= 2pi - arccos(x/R))
=[2*pi- 2 *arccos(x/R)] / (2*pi) = 1- arccos(x/R)/pi
对F(x)求导得密度函数p(x) = 1/( (R^2 - x^2) ^ (1/2) * pi)

极角a服从均匀分布U[0, 2*pi]
设x的分布函数为F(x) = P(X)