一道求统计量分布的题设X1,X2是来自正态分布总体N(0,s^2)的随即样本,求P[(X1+X2)^2/(x1-X2)^
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为了减化记号,用X,Y替代X1,X2.

X,Y为服从N(0,s²)的独立随机变量,二者的联合分布密度函数f(x,y) = e^(-(x²+y²)/(2s²))/(2πs²).

对a > 0,(X+Y)²/(X-Y)² < a² ⇔ (X+Y)² < a²(X-Y)² ⇔ (X+Y-aX+aY)(X+Y+aX-aY) < 0.

设α = arctan((a-1)/(a+1)) = arctan(a)-π/4.

a > 1时有Y/X < (a-1)/(a+1)或Y/X > (a+1)/(a-1).

换用极坐标系,则θ的范围是E(a) = [0,α)∪(π/2-α,π+α)∪(3π/2-α,2π).

0 < a ≤ 1时有Y/X > (a-1)/(a+1)或Y/X < (a+1)/(a-1).

换用极坐标系,则θ的范围是E(a) = (π/2-α,π+α)∪(3π/2-α,2π+α) (α ≤ 0).

P((X+Y)²/(X-Y)² < a²) = ∫∫{D(a)} e^(-(x²+y²)/(2s²))/(2πs²) dxdy

= ∫∫{D(a)} e^(-ρ²/(2s²))/(2πs²) ρdθdρ = ∫{0,+∞} re^(-r²/2)dr · ∫{E(a)} 1/(2π) dθ

= (π+4α)/(2π) = 1/2+2α/π = 2arctan(a)/π.

因此(X+Y)²/(X-Y)²的分布函数为F(x) = 2arctan(√x)/π.

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