※ 引述《Aquarkbrain (腦容量只有夸克)》之銘言：
: 題目：Choose two points on the unit circle randomly. Find the probability density of the length of the chord connecting the two points.
: 不曉得如何下手 謝謝指教
設A(1,0), B(cost, sint), 0 < t < 2pi
d = AB = 2sin(t/2)
其實就是假設AO,BO夾角為t
p.d.f. of t = f(t) = 1/(2pi) (假設t為任一值的機率相同)
P(d≦k) = P(2sin(t/2)≦k), 0 < k ≦ 2
若 2sin(t/2)≦k, 則 t≦2arcsin(k/2)
但此時 0 < t ≦ pi, 2sin(t/2)在(0,2pi)之間對稱於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))
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※ 編輯: cheesesteak (131.179.60.193 美國), 09/28/2020 09:18:38