作者cheesesteak (牛排‧起司)
看板Math
标题Re: [机统] 求p.d.f
时间Mon Sep 28 09:06:54 2020
※ 引述《Aquarkbrain (脑容量只有夸克)》之铭言：
: 题目：Choose two points on the unit circle randomly. Find the probability density of the length of the chord connecting the two points.
: 不晓得如何下手 谢谢指教
设A(1,0), B(cost, sint), 0 < t < 2pi
d = AB = 2sin(t/2)
其实就是假设AO,BO夹角为t
p.d.f. of t = f(t) = 1/(2pi) (假设t为任一值的机率相同)
P(d≦k) = P(2sin(t/2)≦k), 0 < k ≦ 2
若 2sin(t/2)≦k, 则 t≦2arcsin(k/2)
但此时 0 < t ≦ pi, 2sin(t/2)在(0,2pi)之间对称於t=pi
P(2sin(t/2)≦k) = 2P(t≦2arcsin(k/2))
= 2∫f(t) dt, from 0 to 2arcsin(k/2)
= 2arcsin(k/2) / pi
f(k) = d (2arcsin(k/2) / pi) / dk
= (2/pi) * (0.5/sqrt(1-k^2/4))
= 2/(pi*sqrt(4-k^2))
--
※ 发信站: 批踢踢实业坊(ptt.cc), 来自: 131.179.60.193 (美国)
※ 文章网址: https://webptt.com/cn.aspx?n=bbs/Math/M.1601255217.A.8F8.html
※ 编辑: cheesesteak (131.179.60.193 美国), 09/28/2020 09:18:38