mean line segment length

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MathWorld  WolframAlpha

• 半径为$a$的圆内两点的距离的期望 $\frac{4a}\pi$
• 半径为$a$的半圆内两点的距离的期望 $8 (π - 2) a\over π^2$

如何计算呢

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能否加个分类“几何” 就像English area一样?

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半径为$a$的圆上两点的距离期望也是$4a\over\pi$ :
unitsize(2cm);
pair A = (-1,0);
pair C = (1,0);
pair B = dir(60);
draw(circle((0,0),1));
draw(B--A--C);dot(A,red);dot(B,red);
draw(arc((-1,0),.2,0,30));
label("$\theta$",A+(.25,0),dir(40));
label("$2a$",(0,0),S);
$$\frac2\pi \int_0^{\frac{\pi }{2}} 2 a \cos (\theta ) \, d\theta =\frac{4a}\pi$$